It's been a while since I was actively in the PSP scene but a CFW PSP allows you to play your PS1 games without having to re-buy the game on the PSN. Plus you can play PS1 games that have yet to be released on the store.
There are plugins that you can install to take screenshots, play some games in 3d, listen to music while playing games. Another plugin allows you to increase the volume of your psp. I used this one plugin that allowed you to save a battery power by underclocking your psp when you used the hold button. Another program called remotejoy allows you to play your psp on your computer monitor.
There are tons of useful programs and apps for CFW psps, and there is a lot of great homebrew on the PSP. For the first few months of me owning the PSP, I didn't play a single PSP game.
