The Philemon Theory/Analysis
*contains P1 and P2 spoilers*
In the Persona world, there are two deities who watch over mankind. One of them is Philemon, who is born from the best qualities of humanity, and exists as a force of light. The other one is Nyarlathotep, who is born from humanity's worst qualities, and exists as a force of darkness. They are essentially two sides of the same coin.
In Persona 2: Innocent Sin, the two of them have a sort of bet going on, regarding the fate of humanity. Philemon thinks that humans can overcome their bad sides, and become enlightened. Whereas Nyarlathotep thinks that humans are doomed to create chaos, leading to their downfall. Nyarlathotep, being the nasty guy that he is, directly intervenes in human affairs to make sure they create chaos so he wins the bet. Philemon, on the other hand, does not intervene and plays fairly (although his servant Igor helps the P1/2 casts via the Velvet Room).
During Innocent Sin, because of Nyarlathotep's intervention, the entire planet ends up being destroyed (except for Sumaru City, which is now a floating spaceship). To top things off, just as the cast of P2 defeats Nyarlathotep, he pulls the strings of fate once more so that Maya (the female lead) ends up getting stabbed, and dies.
Because Tatsuya and his friends have such strong wills, Philemon says he will be able to reverse everything that happened, by creating a new reality where the main characters never meet each other, and lose all their memories.
In response, Tatsuya punches him in the face (reason #1 of 9999 why he's the best Persona MC), to express his discontent with the fact that Philemon was merely sitting back and "playing fair" while Nyarlathotep had his way with everyone.
After the new reality is created, Philemon is weakened. It must have took a lot of juice to create a new dimesion, because he can no longer appear to the Eternal Punishment cast in his complete physical form. This is the only screenshot I could find showing this, but basically when he shows up, he's either transparent, or a butterfly.
In the new reality, everyone is alive and well, except for two major problems. The first is the fact that Nyarlathotep is still weaving the strings of fate, and trying to destroy the world again. The second is the fact that everyone did forget about the events of Innocent Sin, except for Tatsuya, who's bond with Maya and friends was so strong that not even dimensional-level reality warping could make him forget them.
Anyways, EP goes on with the new cast (and Maya) slowly being exposed to what happened in the other reality, until they confront Nyarlathotep once and for all at the end of the game. After they defeat him and seal him away (although he will never truly 100% disappear), Tatsuya says that he is going to go back to the reality of Innocent Sin to rebuild the world with his friends (who still exist there with their memories). Basically what happens is that the current Tatsuya is his IS mind/soul in his EP body, so when he leaves, the EP Tatsuya that should have been can awaken (who has no memories of IS). Maya, who fell in love with Tatsuya, is sad because they won't ever naturally cross paths again, much less be able to be friends or have the same relationship they had.
Now, this brings us to the theorizing. In the ending cinematic of the game, Maya is seen walking down a street corner. As she waits for the crossing signal, a yellow butterfly (Philemon), appears on the railing. Before it flies off, low and behold, Tatsuya arrives at the crossing on his motorcycle. Ignore the description and comments of the video:
This suggests that Philemon intervened in things, by his own choice, perhaps thinking that it wasn't fair to them to let Nyarlathotep shape their reality as he saw fit. So he worked his magic and made it so that Maya was able to see Tatsuya again. I'm not going to get into the logistics of why Maya smiles at the end (or why Tatsuya doesn't do anything), because it isn't relevant to this theory.
Now, the first time Philemon intervened by creating the new dimension, he was seriously weakened, losing the power to materialize in his physical, humanoid form. So now, after intervening yet a second time so that Maya and Tatsuya connected again, it's pretty safe to deduce that Philemon has completely lost his ability to manifest himself as anything more than his butterfly form.
This is why in P3 he appears a few times as the blue butterfly (notably in cinematics) and in P4 he is hovering over the save points in that form.
So, long story short, that's why Philemon hasn't appeared directly in Persona 3 or 4^^
