First off: I have both systems, and I get a great deal of use out of both. That being said, choosing one or the other definitely comes down to your own gaming preferences. I would break down the pros and cons of each system thusly;
Pros:
Nintendo Brand: Pretty self explanatory. The DS has some pretty damn solid Nintendo exclusive titles, from Zelda: Phantom Hourglass, to Mario Kart DS, to Metroid Prime: Hunters. If you are a Nintendo fan, this one is pretty much a no-brainer
Solid, Deep Library: The DS boasts solid games from most every gaming genre (shooters being the glaring exception). From puzzles (Meteos, Planet Puzzle League), to RPGs (Sonic Chronicles, Chrono Trigger), to adventure (Zelda, Contra 4), the DS is going to give you good options no matter what your favorite type of game is.
Unique and Innovative Games: A number of companies have really figured out how to get the most out of the DS platform, providing gameplay you just aren't going to get anywhere else. The dual screen and touch capabilites give the DS a much different feel than any preceeding handheld. Examples of games that take advantage of this are the Trauma Center games, Hotel Dusk, and Phantom Hourglass.
Cons:
The DS is Lite: Maybe it is just me, but the DS does not lend itself to extended gaming sessions. A lot of its gaming library consists of short session games, so if you are looking for a handheld to play on the crapper or to kill 20 minutes waiting for a bus, the DS is your guy.
Shoot Me: If you are a shooter fan, just walk away now. You have nothing to see here.
Graphics: The DS has decent graphics, but IMO, the PSP just craps all over the DS in this department. Just no contest.
Pros:
PSP spelled RPG: If you are an RPG fan buying a PSP is one of the easiest purchases you will make all year. From Jeanne d'Arc, to the Star Ocean games, to the Final Fantasy games, to the Atlus and NIS titles on the system, the PSP just has a ridiculously good RPG library going for it.
Graphics: It is a PS2 in your hands. Period.
High End Handheld: The PSP is going to give you some great games with high production values and long session play ability. Aside from the previously mentioned RPGs, some other games that come to mind are Ratchet and Clank: Size Matters, Killzone Liberation, and God of War:Chains of Olympus. These are AAA quality PS2 level games on a handheld, like if the DS had Pikmin or Smash Bros.
Cons
Shallow Library: Once you get past the top of the PSP library, there just isn't much depth. The scary part is though, a lot of developers seem to have given up on the PSP, so you better be content with the selection that is out there now.
The Nub: Seriously, who the hell thought that one nub was a good idea? The lack of dual analogs is forgiveable; the stupid ass, useless nub is not.
Price Point: Once you factor in the higher entry cost, the cost of a memory stick, and the *generally* higher price of games, the DS is going to look like a lot better option for your wallet.