Well, since this has been debated already, I guess I'll add my things into it again. Both handhelds support Wi-Fi in a matter of fact. Both can connect to access points, and both can create ad-hoc networks, but the DS, if you wish to use the internet, requires a game cartridge that needs to be bought I believe to use it. The PSP since firmware version 2.0 I think has had a built-in browser with flash capabilities, and as of recent, support for internet radio.
You also forgot some key capabilities as well. Unlike the DS, which is really only meant for portable gaming and some chatting on PictaChat, the PSP is also a DVD player, MP3/WMA player, a photo viewer, a digital camera station, along with an internet radio player, a portable TV, web browser, game system, AND a system capable of doing internet downloads for things such as videos to be played on the PSP.
The PSP does happen to have less battery life than the DS, but this is mainly because of the UMD drive, along with the hardware on PSPs. The UMD drive is very nice, which can read games and videos, and soon to me rumored Audio disks as well. It may be a bit slow, but it can hold quite a bit of data, and with that data, the PSP can deliver some stunning graphics, even if there is the occasional moment where you are overwhelmed with video "jaggies." The DS, while only using memory cards, can hold quite a bit of data too and can do things such as movies, given that there's an app for it and some games, there aren't too many games out there with graphics like the PSP.
Also, the PSP can have custom firmware installed on it. People have literally gotten Windows Vista to run on a PSP, along with Windows games and such using only custom firmware and original PSP hardware. Speaking of firmware, an advantage to the DS is that it seems as though all of the games can run on the original DS software. The PSP has to for some games be updated though, but it can all be done even without an internet connection when needed (via the UMD).
One plus about the DS is that there are more games that people seem to like, like Mario Cart, and more download and play games. The PSP doesn't have many of these though, but it's games are pretty fun.
And of course like Nintendo always is, they make their products cheaper than the other products. Not quality wise, price wise. PSPs are a bit more expensive, but they run for your money, given that you keep the battery charged
Overall in my opinion, the PSP has won. The DS would need some crazy redesigning/reprogramming by Nintendo for it to get in par with the PSP.
