Bladestorm makes an interesting point. I think that there are several factors which have contributed to making G1 the most popular and endearing Transformers series in this franchise's 3-decade long history, and it includes:

Alpha and Omega

G1 came first. Okay, this does seem incredibly unfair to post-G1 series, but there it is. When something comes first then it sets the benchmark for the entire franchise and there's nothing to really compare it to (other than its competitors like GoBots, and Transformers smashed them out of the ball park). Subsequent series, they invariably end up being compared with G1, as G1 has essentially become the "metric" for Transformers. I know that this can often be unfair, but I also think that one cannot deny this reality. It just seems to be the nature of fans to compare things with original sources. Just look at the Deadpool movie; fans have loved this movie because it's far more faithful to the original source material than Deadpool's previous film appearance. But new and different ideas can be successful, but it has the additional hardship of overcoming comparisons with the original. Beast Wars is a good example of this. When this series first came out, it was met with a lot of skepticism from fans, with many crying foul with comments like "trukk not munky." But Beast Wars rose to the occasion and soon proved its worth on its own merit, but in terms of the quality of toys (which has set the standard for Transformers today) and story-telling in the cartoon (which has never been matched on screen; all TF stories that rival BW can only be found in comic books - BW still reigns supreme as the best Transformers screen story ever written... nothing else comes close).

But the whole "G1 as the baseline" thing is compounded when you consider that many subsequent series have based themselves off G1 rather than trying to do something more original.
+ G1 = Optimus Prime and the Autobots vs Megatron and the Decepticons
+ RiD (2001) = Optimus Prime and the Autobots vs Megatron and the Decepticons
+ Armadaverse = Optimus Prime and the Autobots vs Megatron and the Decepticons
+ Movieverse = Optimus Prime and the Autobots vs Megatron and the Decepticons
+ Animated = Optimus Prime and the Autobots vs Megatron and the Decepticons
+ Prime = Optimus Prime and the Autobots vs Megatron and the Decepticons
+ RiD2015 = Bumblebee and the Autobots vs the Decepticons
^One thing that I do admire about RiD2015 is how they're trying to break away from the same tired old formula of "Optimus Prime and the Autobots vs Megatron and the Decepticons!" I wish that they didn't bring Optimus Prime back, but I'm glad that they've made him subordinate to Bumblebee rather the usurping him. And I'm glad that they didn't just bring Megatron back (in Season 1 at least, I haven't finished watching S2 yet so please no spoilers, thanks). I quite like Steeljaw as the de facto leader of the Earthbound Decepticons; he kinda reminds me of IDW Soundwave, albeit more sinister. I was disappointed that Michael Bay just brought Megatron back (as Galvatron) in AoE... no imagination. If people want new Transformers series to leave the shadow of G1, then that series really needs to stop trying to emulate G1. This is another reason why Beast Wars was so awesome; they weren't trying to be like G1.

The Toys

The toys captured our imagination and fascination. While many G1 toys have become outdated by modern standards, they were mostly excellent toys for their time. The first year of Transformers become the highest grossing debut toyline, raking in US$1,400,000 in the US alone (which was a lot of money in 1984!). e.g. MP Soundwave's transformation is almost the same as G1 Soundwave's, but that just goes to show how well made G1 Soundwave was. MP Soundwave didn't deviate much from the original design because it didn't need to! You can't beat perfection. G1 Soundwave was a perfect toy by 1980s standards, and MP Soundwave is a perfect toy by 2010s standards. We know that the G2 toy line didn't fare as well because it failed to capture this same imagination. Early toys were just redecoed G1 toys, and while G2 did give us some really great toys like the Combat Heroes, Laser Rods etc., we did also get some really rubbish figures like the Power Masters etc. Beast Wars was the next line to recapture our imagination, as proven by impressive toy sales, and Car Robot/RiD2001 sold like wildfire with toys flying off shelves months before the show started. And we know that Car Robot went on to inspire Binaltech/Alternators and Masterpiece, so while that series has ended, its legacy lives on. But some of the post-CR/RiD series arguably failed to capture our imaginations at that same level. Armada and Energon saw a lot of gimmick dominated toys (Cybertron was a lot better though ).

The Great Shift
Since 2007 we've seen Hasbro shift towards Transformers based on animation models instead of animation models being based on toys (as was the case with most pre-07 TFs). This was understandable for the live action movie toys, but Hasbro has since applied the same principle with other lines, namely Animated, Prime and the current Robots In Disguise series. The issue that I have with this is that animators and artists don't know how to design toys. They're not engineers. So it's been up to HasTak to "translate" animation models into transformable action figures, and quite frankly, it's a tough job. With most pre-2007 Transformers, HasTak's designers never had to worry about this, they could just focus on making the toys for the sake of being toys, not for the sake of emulating a screen model. There was far more creative freedom; let the animators 'translate' the toys as screen characters. This worked brilliantly in Beast Wars. When I look at all my Animated, Prime and RiD2015 toys, none of them strike me as being particularly excellent. Just good at best, but on the whole, I find these lines to be rather mediocre. There's nothing in these lines that I find on par with figures like G1 Soundwave, Ultra Optimus Primal, Mach Alert etc. where I just think, "That's a really clever toy!" It's not capturing my imagination and fascination at the same level as previous figures.

The Canon

One thing that made G1 stand out was the quality of story telling. Even with the G1 cartoon telling a story at a more superficial level (the Transformers were more caricatures than characters) and camp factor, the show didn't talk down to the audience. This is something which G1 voice actor Michael McConnohie (Tracks, Cosmos) has spoken about in interviews. Kids hate being treated as idiots. Many other toy franchise stories in the 80s had a condescending tone, but not Transformers. Transformers was more than happy to use mature language, which was littered throughout the tech specs, comics, cartoon etc. I learnt a lot of new words through my exposure to Transformers. G1 writers ensured that their characters spoke properly, demonstrating basic standards of language. This is incredibly important when it comes to children's literature too. I remember first watching Transformers on TV and my mum came into the lounge room and tried to convince me and my brother to "turn that rubbish off," but when she heard the characters speaking, she instantly noticed that they were well spoken. She then allowed us to watch Transformers, but told us to be mindful of the way that they spoke in order to pick up new vocabulary and grammar. This is what educators call the "hidden curriculum" in literature. The G1 cartoon occasionally hit on some really heavy themes too. McConnohie once used "The God Gambit" as an example; an episode where there's a planet whose gods resemble Cybertronians, and when the Cybertronians arrive they initially worship them as gods, and the Decepticons happily exploit this. This episode explores how religion can be corrupted as a means of controlling the masses and examines what people are willing to do in the name of religion. The Autobots on the other hand demonstrate religious tolerance - insisting that they're not gods and being appalled at seeing the Decepticons abuse the indigenous people's beliefs.

The G1 comic continuity, of course, was written at a significantly higher level than the cartoon, where the Transformers were actually treated as characters, and many stories were written in ongoing arcs rather than stand alone episodes (although they were plenty of those too). This is something IDW is doing now. And IDW has taken many cues from the Marvel Comics, such as the Wreckers, Decepticon Mayhem Attack Squad, ancient Cybertron being a corrupted republic (although this republic was far more sinister in IDW; in Marvel they were just incredibly arrogant and naive) etc.

Quirky Characters!
Here's something that is arguably lacking in more recent Transformers series -- individual quirks. This was something that G1 established through its toy continuity, much of which was picked up in other continuities, but just looking at the original toy canon, we can see that while many Transformers had awesome abilities, they also had bizarre quirks. This helped to flesh them out as being interesting characters, and not just, "butt kicking robots!"
e.g.
* Bumblebee; while physically weak, his inconspicuous (i.e. not an attention-drawing sports/race car!) alt mode made him the ideal spy. There's a scene in Captain America Civil War where Cap, Bucky & Falcon are travelling in a VW Beetle to avoid drawing attention to themselves. But at the same time, Bumblebee is also desperate to prove his worth to his fellow Autobots.
* Soundwave; uses his incredible eavesdropping ability to blackmail his fellow Decepticons, who all hate him.
* Mirage and Thundercracker; moral ambiguity.
* Bluestreak; former refugee/asylum seeker whose cheerful exterior masks a tormented past.
* Trailbreaker; riddled with guilt as the most fuel consumptive Autobot.
* Breakdown; paranoid. WHO SAID THAT?!
* Dead End; emo.
* Motormaster; hate-mongering hate ball who is hated by his fellow Stunticons. This causes Menasor to be one big mentally messed up gestalt who finds it hard to think.
* Hook; The Decepticon equivalent of Hannibal Lecter.
* Groove; pacifist (IDW Megatron is currently more like Groove should be!)
* Blades; violent thug and all round adrenaline junkie. Surprisingly Blades and Groove are able to set their differences aside and work together professionally as components of Defensor, as his mind isn't fragmented like Menasor or Devastator's.
* Cosmos; lonely and desperate for companionship.
* Snarl; lonely and that's just the way he likes it. Hates the company of others.
* Shockwave; logic fiend.
...etc etc.

And we know that IDW comics have been feeding off the original source material from the G1 tech specs, Thundercracker being a notable example. IDW also saw the first canonical appearance of Hasbro's (not Takara's) Overlord, as well as the longest canonical appearance that Hubcap's ever had (although in other cases, completely deviating from it; e.g. Swerve, Tailgate etc.).

Not too many post-G1 toy lines have 'seeded' themselves with so much rich story potential via their toy continuity such as their tech specs. One of the brilliant things that Hasbro did with Transformers and G.I. Joe was to get Marvel comic writers to create characterisations for the toys. Larry Hama worked on GI Joe, and Bob Budiansky worked on a lot of the profiles for the Transformers. IMO few post-G1 TF lines have well written tech specs; those that do were Beast Machines and Transformers Cybertron.

Longevity

The original Transformers series (G1) ran for 8.5 consecutive years. When I think about why Beast Wars, despite its awesomeness, hasn't achieved the same level of fan endearment as G1, I think one contributing factor was its significantly shorter lifespan. Beast Wars only ran for 3 years (1996-99) before Hasbro cancelled it, and in the opinion of many fans (myself included), prematurely so. This is why the latter half of Beast Wars S3 feels so rushed, because it was. Writers had to wrap up all of these loose story ends because Hasbro was cancelling the line. But this gave it a relatively shorter span of time to sink into the fandom's psyche.

Subsequent series haven't fared much better. Beast Machines and Car Robot only lasted about a year each, and others have barely lasted beyond 3 years.
* Unicron Trilogy = 2003~06 (well, it started at the tail end of 2002, but most stores didn't get these toys until 03)
* Animated = 2008~09
* Prime = 2011~13
* Robots In Disguise = 2015~16
So Hasbro's now really into constantly rebooting the franchise on a fairly regular basis. The rationale being that it keeps the line fresh, but the downside is that it makes it difficult to build a strong fan base. Also, these lines are often used to fill the gaps between the live action movies, then a new movie comes out and the mainstream kids line gets rebooted. The live action movie franchise started 9 years ago, but bear in mind that these are not consecutive years; there are 2~3 year breaks between each film. Basically, there's not much in the way of long running continuity with post-G1 Transformers series.

Now, G1's longevity is even longer when you consider the G1 Renaissance, which may have started as early as 1996.
* 1996-99: Beast Wars & Beast Machines. As we know, the Beast Wars cartoon established itself as being in the same continuity family as G1, enhancing G1. Beast Wars also gave us a few characters that were explicitly stated as being the actual G1 characters themselves, such as Grimlock etc. Beast Machines continued from that and even started retroactively introducing new characters into G1, namely Obsidian and Strika.
* 1997: Machine Wars. All of them being new versions of G1 characters.
* 2002-08: exclusive G1 redecoed characters. Ichikawa Hirofumi's work on creating new characters for the eHobby and convention exclusive toys served to further expand G1, giving us characters like Road Rage, Tiger Track, Clampdown etc.
* Binaltech; again, more of Ichikawa's work to further expand G1, and also retroactively explaining things from G1 (e.g. Prowl II etc.).
* Masterpiece. New characters like Exhaust and Loud Pedal continue to expand the G1 universe.
* Classicsverse. While it's mostly upgrade of previous Transformers, Classicsverse has also given us a fair number of new characters too, including Drift, Windblade, Rook, Off Road, Victorion etc.
...in other words, G1 has continued to grow, it arguably has never stopped. One might argue that G1 never died off, but rather, evolved into becoming these "Renaissance" series. Much like dinosaurs evolving into birds.