You've gotten a few perspectives but i kinda wanted to offer my perspective on it anyway. I love smt and persona so grain of salt i guess. I'm gonna try to offer a detailed breakdown in case you're into that sort of thing.
SMT games are ones where you fight mythological figures. In many games, enemies are named after mythological figures, but in SMT they are the actual figures themselves. It takes place in a universe where Christian God is dead, and so all these other mythological figures from religions past are trying to fill in the gigantic power gap. At the end of pretty much every game, you fight some religion's primary religious figure. One of em has you fight Yahweh, for example. It's pretty fuckin' sweet.
The series' story has honestly generally uninteresting characters, tending to put its focus more in building very interesting settings and plots. They tend to focus on the ideas of law/chaos, which are regulation vs freedom. Both are presented as undesirable, with neutrality being the true ending in pretty much every instance. Law results in a world where humanity surrenders its power to an overlord who decides everything for them, while in the Chaos route, all but the strongest die at the hands of these newly unleashed gods.
The games' combat is turn-based, and focuses on the fact that every move has an elemental alignment. Every character and enemy reduces damage from some elements, but takes extra damage from others. Usually, if you or the enemy lands an attack with an element the target is weak to, an extra action is added. The combat flow is largely about trying to squeeze in as many actions as possible, and give the enemy as few actions as possible. There is also a strong focus on using status ailments and buffing/debuffing stats, which distinguishes its fights from those of many other RPGs.
In addition to this, the game has a Pokémon-like system by which you can acquire the enemies you fight, adding them to your party. You can then combine these enemies into stronger ones. There's a focus on leveling up one god and getting the best skills it can learn, then passing it on to a higher level god with better stat spreads and resistances. It's effectively a weapon-crafting system, but the crafting is on the scale of characters.
All in all, the SMT games have a lot of elements of traditional JRPGs, but with a rock-paper-scissors element to the combat,
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At some point, the SMT games' timeline splits and on one branch you get SMT and on another you get Persona. A simplification, but there you go.
In Persona, the focus has changed somewhat. I'll mainly be talking about 3/4 since I did not play 1/2 and am led to understand they differ signifcantly, being a lot more like SMT. In Persona games, you are a high school student. Half of the game is about living a life as an ordinary student, while the other half is about fighting monsters in a dungeon, generally secretly.
Persona generally has less interesting settings than SMT, but more fleshed-out characters. Almost every named NPC will have 11 or more events that flesh out their characters, usually focusing on their fears and the areas they need to grow in maturity in. Few video games focus on a lot of the issues the NPCs go through, and I think it's this aspect that makes the characters particularly interesting. In Persona 4, for example, there is a strong emphasis on a pair of best friends who, although they do sincerely love each other as best friends, often feel jealousy at times of each others' strengths. They end up confronting these feelings and admitting that they do feel them, but try to get over them as best they can. Usually the "good guys" in games are kind of perfect, and those sorts of feelings would make a character evil, but it's very much presented in a way that, yeah, human beings feel bad emotions sometimes, even if they are generally good. There is nuance to the characterizations of the NPCs in a way that is rare for video games and media in general.
In the "normal world" segments, you can choose how to spend your days. These vary from just hanging out with friends, to working part-time jobs, to joining after school clubs, and more. Most of these have ramifications with the dungeon crawling segments, generally making it easier in some regard. If you become good friends with a party member, their combat abilities get enhanced, for example. They'll be able to do stuff like crit against an enemy more frequently, or survive an attack that would kill them with 1 HP. That sort of thing.
There's actually a lot of systems involved in this segment, and although they're pretty intuitive, there is some depth and balance to them. To give one quick example, every character you can hang out with is represented by a Tarot Arcana. Additionally, every god you can fuse is also represented by a Tarot Arcana. By becoming better friends with Chariot, you can enhance the powers of Chariot gods. And if you have a Chariot god in your "party", your relationship grows more quickly with the Chariot character. However, you can only hold between 6 and 12 gods in your party at any given time, and some of the most powerful ones are in Arcana that don't actually quicken this aspect of the game. So there is a balancing act in determining whether you want to make the combat easier with stronger gods or you want to focus on your friendships and take potentially weaker gods to make your relationships grow faster. That's just one system but there's several more that add nuance to the relationship system. It helps to add to the balancing act that the games hinge on, where you also generally want to try to go into dungeons as infrequently as possible, so as to have more time to make friends and junk.
Persona retains a large number of systems from SMT, but generally it reduces the number of elements and makes fights easier to solve via overlevelling than they would be in SMT, where overlevelling is generally not very useful at all. The game is just easier overall. Another difference is that SMT doesn't have a huuuuge amount of straight up dungeon crawling, which Persona's dungeons are entirely just floor after floor after floor.
Also noteworth is that both series have some of the best music in games, and an insane variety, ranging from
rock/metal to
hip-hop to
pop (these are all overworld themes), and the Persona series has some of the best UI design I've ever seen in a game.
Whew. That was really long-winded