FPSE is the best PS1 emulator from my point of view, when PSX4droid came out, i was so happy that i got it the day it came out, but my hopes and dreams were shattered when i saw that almost every game i threw at it played and sounded like crap, sure i could OC my phone to the max and turn the sound off in game to get a good 30+ FPS in most games, but for me it wasn't enough, and then....FPSE came out, after buying PSX4droid, there was no way i was going to spend money on another Playstation emulator, till i saw videos of FPSE in action, it was like a ray of light shot out of my monitor and punched me in the face, but it didn't hurt, in fact it opened my eyes, and i went and i got FPSE, and i NEVER looked back, it ran every game i threw at it WITH SOUND at 50-60 FPS
TLDR; PSX4droid,the dev took everyone's money and booked it and just made the app free, so if you want an emulator that runs like crap and has no support from the dev but it's free, then PSX4droid is for you, but if you want an emulator that runs everything almost perfectly, has support, updates like crazy and just keeps getting better and better but costs money, then FPSE is the one for you
Hope this helps!