I prefer the psp because the graphics are better and it has alot of awesome games (crisis core, disgaea etc), but as has been mentioned, it does have a pretty short battery life. I also think that including a touch screen in a console was a stupid idea as was the dual screen thing.
DS. Not only is it semi-backwards compatible (can play GBA games but not GBC), but it has cool new touchscreen technology and innovative ways of solving puzzles (People who've played Phantom Hourglass should know what I mean)
I have a DS but the PSP is very shiny and nice. I'm also a big Sony supporter, but I'd have to say DS wins it for this one. However, I'm all for the Playstations.