Though I do not own one I'm inclined to agree with phillyX that if homebrew software is your thing, the PSP is a superior device. The DS has its own
vibrant homebrew scene but if you care a lot about technical specs, the PSP is the handheld for you...
HOWEVER. If we are to compare existing game libraries, the DS wins by a long way. I can think of only a handful of games that interest me on the PSP (
LocoRoco,
Echochrome and
Crush) while the DS has things like
Hotel Dusk and
Another Code in addition to old favourites like Mario, Zelda and Tetris. The stylus and touch-screen make a great interface when used properly.
Trauma Center: Under the Knife, for example, is one of the most fascinating and original games I've played in ages. Basically you get to be a surgeon with a stylus.
But, if I had to give only
one reason to recommend a DS to an adventure enthusiast it would be:
Phoenix Wright: Ace Attorney.