The PSP is an excellent multimedia device first, decent games system second. The DS is purely a game device and an excellent one at that.
Graphics: The PSP is nearly the equivalent of a handheld PS2, while the DS is a little more powerful than a Nintendo 64. Both can do 3D, but the PSP is hands-down more powerful. The DS has a few nice 3D games, like Metroid Prime Hunters, but they don't even come close to a game like God of War:CoO for the PSP.
Screens: The PSP has a lovely large screen that is excellent for watching video, while the DS has two separate screens of lower resolution.
Case: The PSP has a glossy, exposed surface that is easily marked with dirt and fingerprints and every PSP owner I know immediately buys a case for it. The DS folds in a clam-shell design and has no need for one.
Controls: The DS is generally more ergonomic than the PSP's nub. Most DS games feature multiple control methods, while many PSP games feature compromises to it's controller design. (since much of the PSPs library consists of PS2 games designed with two analog sticks, this can be problematic).
Usage: The DS has excellent battery life compared to the PSP. The PSP has multimedia features that the DS cannot match...regretfully, the iPod Touch CAN, which is why all my friends who bought PSPs tend to play on the DS and do everything else on an iPod, instead.
Software: The PSP's library, while not anemic, features a LOT of PS2 ports. The problem is that most of them are direct ports with little difference from the original game. The controls of the PSP often suffer from the translation, and many of these games weren't designed for a 'travel' style of gaming (i.e. for playing in short bursts or interrupted periods). The DS library is much larger, though there is a large number of titles for younger players. However, that doesn't mean there isn't support for more hardcore players...in fact, the DS is the home of some of the best 2D games in the last decade, such as the Castlevania games and Contra 4. The DS also has a huge library of RPGs, especially the well-done 'reimagining' of Final Fantasy III and IV. Also, the DS can play GBA cartridges, which is kind of nice, too.
Online: The PSP wins this hands-down. The DS support for wireless is adequate, but anemic...a common Nintendo trait. The PSP allows for much better online experiences. Both have some capacity to connect to their 'big brothers', but Sony's support in this area is much stronger...if you have a PS3. However, the DS has some great 'local wireless' features, and many games support multiplayer on the local level very well (from the old Pokemon match to playing Contra 4 cooperatively).
My personal recommendation would be to see what kind of games you like and how much you're willing to spend. I personally think the DS is better for what I want to play, and have four in our house. I play mine daily on the train. Current games: Apollo Justice, Warhammer 40000 and Contra 4. But make sure you get the DS-Lite, not the DS. It's better in almost every way. And if memory serves, the Guitar Hero On Tour bundle for the DS comes out on Sunday.