Art-Style Dissonance: Starting with 3, the series became much more brightly colored in general, including quirky non-human sidekicks (Especially Teddie). However, the game is still not afraid to mention characters dying (in sometimes gruesome ways), show enemies that heavily resemble naked women in disturbing contexts, or even touch upon themes like impending death and apocalypse cults. In fact, the series has arguably been getting more graphic as it goes on, with all the games from Persona 3 onward being rated "M", whereas the first three only got a "T" rating. (The first also got a K-A rating, but that was a heavily Macekred version.)
Special mention goes to Persona Q, which has a chibified art style in the palette of Persona 3 and 4... and a lead who is the ghost of a person with a terminal illness, with most of the bosses representing the lead up to death.
Broken Base: With the main series, due to its popularity overshadowing it and everything else Atlus makes. There are those who actively dislike the spin-off and its fanbase, viewing them as Johnny-come-latelies who can only enjoy watered-down gameplay and thematic compromise. Simultaneously, some fans are only fans of the spin-off and dislike the main series for its stagnant, recycled "order vs. chaos and both are jerks" plots, flatter characters that contrast heavily with the spin-off's heavily character-focused narratives, and its archaic difficulty and gameplay. There are those who enjoy both, but the two series are different enough from one another that there's less overlap than one would think.
The Persona fanbase itself has suffered its own break, mostly between the fans of the first two games and the fans who jumped on to the series at or after 3. The first two were much more heavily tied to the original Shin Megami Tensei franchise in plot, style, and gameplay than those after 3, where the Dating Sim elements began to enter the mix and the Tarot Motifs became much more prominent, and possessed shared elements that have, if not totally been abandoned, then at least somewhat sidelined following 3. (Philemon and Nyarlathotep being Put on a Bus is a common example.)
Between the releases of Persona 4 and 5, there was the substantial-and-growing portion of the fanbase that was simply tired to the bones of Atlus constantly adding spin-offsto the spin-off for a quick buck and wanted the series to progress, given that Persona 4 came over half-a-decade and two console generations before the next mainline Persona game. Even after Persona 5 was finally announced, Persona 4: Dancing All Night still added fuel to the fire even as it addresses those fans' core concerns.
To put it in perspective: Atlus hosted an official booth at Anime Expo 2014. The only material they promoted was Persona 3, 4, their spinoffs, and nothing else, not even other Atlus IPs. And then something similar happened again during the announcement for the series's 20th anniversary where the earlier titles were treated as mere footnotes while 3 and 4 got loads of promotion and references.
Continuity Lockout: Averted - every game is essentially a standalone title. If a player wants to start at Persona 4, for instance, they have nothing to lose but a few cool Mythology Gags. The only mainline game that is a true sequel is Persona 2: Eternal Punishment, and even there much time is devoted to explaining the events of Persona 2: Innocent Sin for newcomers. In fact, since Innocent Sinnever officially left Japan until the PSP remake,Eternal Punishment was a sterling example of avoiding this. The idea of a completely different adventure that had happened off-screen, leading up to the innocent sin, actually intrigued a lot of North American players.
First Installment Wins: Averted - The first game was only a cult classic at best, and has not fully enjoyed the gains of the breakaway success of the series following 3 and 4, partially due to their drastic differences is story, style, and tone.
That being said, this could be an example of this for the Megaten multiverse as a whole. Revelations: Persona was the first Megaten game to get a western release, and the Persona series as a whole is much more popular in the west then the Shin Megami Tensei series that birthed it
More Popular Spin-off: The Persona games (specifically the games from Persona 3 onward, especially 4) are the most popular commercially and critically acclaimed about of the whole Shin Megami Tensei franchise.
Never Live It Down: The 10 month delay of Persona 4: Arena in Europe gets brought up every time the series is discussed. Regardless of the change in circumstances for the franchise since thennote Namely, Sega's 2013 purchase of Atlus meaning that Sega UK are now publishing Atlus games in the region as opposed to different publishers on a game-by-game basis, the EU fanbase still holds a grudge. Other titles also coming out late didn't help.
Sequel Displacement: How many people knew of Persona before the third or fourth games? For that matter, it's rare to meet a Western Shin Megami Tensei fan that wasn't introduced to the series by those two games.