Well, I own both handheld consoles so...

PSP
Can play music and the screen is lovely when watching 16:9 movies, series or animes. The 16:9 screen is superb when 16:9 videos come into play. I cannot express greatness of the screen. Watching movies on a handheld has never been better with this screen. Just use Xvid PSP to convert the movies into PSP-playable movies. Quality of movies are great.
Games are like your normal PS2 games, nothing much interesting there. The games now are quite boring and nothing interesting are around.
Battery life is a problem.
Good browser. Supports WPA, while the DS only supports WEP. WPA is better than WEP.
I normally use PSP to watch movies since the games are... well... erm...
It tries to be an all-rounder and it succeeded.

Nintendo DS
The games on this handheld are awesome. At first I thought Mario Kart DS was boring but once I played it with my friend, we went hardcore for 4 to 5 hours playing Mario Kart DS until my thumb hurts from pressing the A button. The touchscreen really works wonders. Games with intensive touchscreen usage are great (though there are some which are terrible). I really love the games on this system despite its graphics are not as good as the PSP.
Able to watch movies and listen to music using flashcarts albeit movies are not shown superbly on the DS compared to the DS because of separate screens. Hard to read subtitles on movies or animes.
Battery life is great.
Have to buy your own internet browser, or get an Ez 3 in 1.
Depending on flashcarts, you can have real-time saves, real-time guides and stuffs like that. Amazing, isn't it
It works as a gaming handheld and yeah it rocks at the games, even first-person shooter feels amazing on the DS (albeit the graphics are... well... acceptable).
Love the homebrew applications on the DS. Personal favourite: Comic Book Viewer DS. Just amazing.

Opinion
I personally prefer the Nintendo DS when it comes to games. The games are so much fun, Mario Kart DS is one of my favourites. Pokemon is great too. Adventure games such as Final Fantasy III and IV are awesome. More and more RPGs are coming to the DS, so if you love RPGs, better get a DS. Strategy games work wonders thanks to the touch screen. Though movies and mp3s can be played through flashcarts, their quality is not as good as the PSP. Internet browser easier to use thanks to touch screen. Easier to type stuffs.

If you prefer an all-rounder handheld, get a PSP. Its games aren't as interesting as the DS but there are still some which are great to play such as Final Fantasy 7: Crisis Core and Metal Gear Acid. Currently playing Key of Heaven. Love the internet browser but typing is excruciatingly cumbersome.