Emulation is acceptable up to the complexity of SNES+expansion chips (and the PSP is still competitive as a portable emulation machine, not in absolute power but as a platforms/price ratio)
http://wololo.net/emulators-for-the-psp-ps-vita-the-ultimate-download-list/
Of course, it can also play all PSP and many PS1 games, music, photos, and videos!
You can also install plugins to enhance software - say, turning off the screen when you slide the power switch in the lock direction? (slow) savestates? soft reset? continuously variable brightness above and under the official limit? screenshots everywhere and anytime?! sorting apps into folders? audio streaming fix for PS1 mode?
As for non-utility homebrew apps, I'm not too familiar with them -- I've tried the joke "Mario Kart PSP", a Minecraft clone and not much else... the PSP was the only really successful competitor to Nintendo handhelds, was powerful for its time and price and it was hacked very early, which definitely reflected in number of homebrews -- but like for DS, a not that small number of them are now hard to find...