What the PSP has:
A. Tons of homebrew. It can emulate everything from Atari to GBA with near-perfection. PS1 games are emulated now and they run great. And there's an N64 emulator that in the next couple months will be running at great speeds (right now Mario 64 is just barely playable, but it's only a couple months in development)
B. Aside from the emulators, there is also a lot of custom games, including one called "SnakeSP". If you have a PSP, I highly recommend you get that game, it's VERY good looking, free, and fun.
C. There is also a ton of customization. Just about everything in the PSP's XMB has been hacked, including fonts, waves, wallpapers, icons, gameboots, removal of the PSP logo, etc.
D. There is now custom firmware which allows people to play games on their terms, as well as add other features such as playing your MP3s ingame, taking screenshots, and turning on USB whenever.
E. There is a lot of "functional" homebrew, such as IR Remote software, VNC, links web browser, text/pdf readers, manga readers, wifi sniffers, FTP/HTTP servers, shoutcast internet radio players, streaming video players, various audio/music related software, etc.
F. It plays MP3s, WMAs, ATRAC3/plus, AVC (h264) video, pictures, and more. It has GPS capabilities, and has (admittedly low-quality) a camera addon.
G. It still has quite a bit of fun and excellent games, including the ultimate puzzle game since Tetris: Lumines. I swear to God I would buy a PSP just for that game alone. Wipeout (a new one is coming soon), Ridge Racer, Lumines, Every Extend Extra, MGS, plus more I want to get.
Not to mention it's one of the only handhelds in the history of the gameboy to actually hold its own in the market. I won't argue that the DS has a much greater market share, but to say the PSP hasn't made an influence, or at least gave the DS a run for it's money? You'd be lying to yourself.
It pretty much comes down to the DS being strictly a gaming device, while the PSP is an all-around entertainment device, which I feel fills an empty void in mobile entertainment. It does everything, and it does it all well.