The PSP is better graphics wise and has more games for serious gamers and the DS has more kid friendly games. That being said the PSP is struggling and if your thinking about buying one I would stay away from the PSP and get the DS.
While the PSP may continue, support from game developers is dropping and they may discontinue the PSP or come out with a new one altogether.
The DS though has a lot of bad games, mostly ones made by Nintendo like Mario, Zelda, etc seem to do well with a few third party games doing well.
So depending on what your looking for either or neither may be your option. You could always skip both if you have a good gaming phone like ones from Apple or Android.