Two versions of Persona 1 are available to play in English. The first is the 1996 PS1 original, which was localized in the West as Revelations Persona. The second is the 2009 port for the PSP, which is titled Persona 1.

TL; DR: Play the PSP port.

The PS1 Revelations Persona features an infamously bad localization, with the setting and cast changed from Japan to America. Character dialogue, Persona names, and spell names are mangled, and half of the game is missing due to the Snow Queen Quest being cut from the localization.

On the other hand, the PSP port has a much better translation, retaining the characters' original names and the Japanese setting. and you can play the entirety of the game. It is enhanced with quality of life features such as fast forwarding battles and higher quality cutscenes. It also features a redone soundtrack with entirely new vocal tracks by Shoji Meguro to be more in line with the music styles of the newer Persona games. But some (including myself) think that the PSP music is inferior to the original PS1.

There are two recommended patches for a better game experience: the Lower Encounter Rate Mod and the Restored PSX OST Mod. In the second link, there is an option to download both mods in a single patch. The Restored OST mod only restores the soundtrack and not the original sound effects, such as the footsteps in cutscenes and negotiation sound effects, but it’s good as it is.

Previously, a reason why people might prefer the PS1 to the PSP version of Persona 1 is the presence of Hidehito Aoki’s original soundtrack, which gave the game a distinctive surreal atmosphere. But with the release of the Restored OST patch for the PSP, there’s no need for that. More on that below.

Now, you should play Revelations Persona only if you really want to know what the original game was like, or if you want to experience its botched localization for yourself.