It gets asked every so often. Here's the short rundown.
Persona 3 FES has:
- Three difficulty levels - Easy, Normal, Hard.
- Only the male protagonist
- No direct party control; you must issue tactics to your party members.
- Animated cutscenes
- Proper in game cutscenes
- Stricter gameplay ruleset than Persona 3 Portable.
- Fatigue system in Tartarus. Your party members will eventually get tired after so much exploration and dungeoning.
- Multiple weapons for your protagonist
- The Answer epilogue after you beat the main game
- Available on the PS2 and PS3
Persona 3 Portable
- Five difficulty levels - Beginner, Easy, Normal, Hard, Maniac.
- A male and female protagonist route, with different social links.
- Direct party control, in contrast to issuing tactics like FES.
- Numerous minor UI improvements from Persona 4
- Looser gameplay ruleset based upon Persona 4's system and mechanics
- No fatigue system in Tartarus; You can stay in there as long as you wish and have the SP to do so.
- Single weapon for the protagonists; no weapon switching.
- No animated cutscenes
- No epilogue; The Answer is not included
- Visual Novel like cutscenes and presentation.
- Available on the PSP and Vita.
Most people say to start with Persona 3 FES due to it's better presentation, but some argue in favor of Persona 3 Portable because of the female protagonist and direct control. You can't really go wrong with either one tho. Choice is up to you.