Since purchasing my PSP early this (now past) summer I have noticed something about nearly all of the games I own. Most of them are simply console experiences shrunk to fit on a PSP screen. The play style, save structure, and overall experience is exactly what you might find on your PS2. While this sounds fantastic, it is quite clearly the biggest failure of the PSP.
Arguably the most important role of any portable game system is providing a portable game experience. This may sound obvious, but it is pretty clear Sony wasn't considering this when they developed the PSP. The games they promote and encourage, such as God of War: Chains of Olympus which came as a pack-in with my PSP is by no means a portable game. It doesn't allow for quick and easy saves and doesn't give me the ability to play it in short bursts and pick it up and play it again later. This problem can even be extrapolated to describe the PSP hardware itself.
The PSP hardware is disc-based thus effectively ruining any sort of battery life the system might have and only allowing for somewhere around four hours of continuous play. Furthermore, just getting the system started and into an actual game takes entirely too long. After two (or more) minutes the game is finally available for play.
The DS suffers from none of these issues. Most games are designed to be highly portable, cartridges ensure little-to-no loading time, and battery life is phenomenal. Just these factors alone ensure that if you want to have a game system with you to play outside of your home, the DS is the one you are choosing.
PSPs are technologically fantastic machines that do some really great stuff, especially when paired with the PS3, hell even a lot of the games on the system are quite good, but it utterly fails at doing what it is supposed to do; be a portable gaming machine.