I currently own a PSP but often play my friend's DS too, from my experience I can say that each has their strength and weaknesses :
1. Game
the DS's game library is Huge! it has so many good games (and the touch screen is so fun to use), while PSP's game is sometimes just a port of the original PS / PS2 game. however, although in a small quantity, PSP has really good original games (with AWESOME graphics), like MGS portable ops, Loco Roco, lumines, GTA Vice city stories etc. And firmware 3 and above can use PS emulator too. And if you like RPG FF7 crisis core is coming up
2. multimedia capabilities
I'll have to say PSP wins, hands down. the widescreen display is beautiful, it supports mp3, mp4, wav, flash (usually I can fit 3-4 episodes of hello morning to a 2gb memory stick). the speaker is still a bit better than DS (IMO) though using an earphone is reccomended. yes the DS can play mp3 & movies too (I've tried it) but the small display is a problem.
3. size
PSP is really sexy, but a bit bulky (and heavy), while the DS(esp. DS lite) is compact and lightweight.
4. battery life
my PSP can run about 4-6 hours nonstop (depending on the brightness setting) and only takes 1-2 hours to charge. I don't really know about the DS though.
5. internet browsing
I'll say PSP is better, though it still suck at browsing, since it only has a 2mb cache to store images etc (everytime I open this forum from my PSP it has trouble loading the image because there's not enough storage space)
6. price
PSP is more pricey, not to mention the price for those memory sticks >_< but It's worth it. DS is cheaper.
7. loading time
this is a big issue, because psp takes some time lo load from the umd, while DS loads really fast because it's using card. however my psp plays pirated game from the memory stick so it only takes 1-2 seconds to load, heheh
Too bad that neither the PSP or the DS-Lite allow you to run pirated/downloaded games. I'm not looking forward to buying $40 games...
actually both can run pirated games
for psp check out psp-hacks.com. the DS uses something called supercard, so you put your rom in it, and just insert the card like an original game. the DS has an advantage here because DS roms has an average of 100mb size (most are smaller). while PSP games can go up to 1,7GB in size (like gitaroo man lives) though averagely it's 200-600mb per game. that's why if you plan to have your psp play pirated game it's better to buy 4gb memory stick with it(I have 4gb for games and i bought an extra 2gb stick for my video files).
conclusion:
again, the desicion is up to you, but I recommend you to try them both first, play a few games on each, and decide which one you like better
they're both great game system.
hope this helps, sorry if it's too long :p