I have both the DS and PSP. The DS has over twice the amount of sales the PSP does in hardware (130 million vs 60 million). I like some of the games on the DS like New Super Mario Bros., Mario Kart DS, Pokemon Diamond, etc. However, I don't see why so many people have it compared to the PSP. I like the PSP more because it provides PS1 classics(which is awesome). So the main question is: Why do you like the DS?

Nintendo games is the biggest reason for me. The touch screen is fun too Like Elite Beat Agents and Hotel Dusk. Also, all the JRPG's it now has even though I will never get them all. Oh, and Nintendo games.....

It has the World Ends With You, and Trauma Center, pokemon,  though I don't like pokemon like I used to

But I like: Wipeout, and Crisis Core on PSP, with ps1 classics

so for me it's even

I don't really like very many games on either

Why wouldn't I like the DS?

The games are awesome and I don't really care that it's not a media device since I already have a better media device (the iPhone) than it and the PSP.

I like Pokemon. That's why I like the DS.

I like the DS because it gives me access to the latest and greatest Nintendo games. Not to mention the countless JRPGs available.

$170 for a handheld is way too much, and it doesn't have any games I want to play.
I got my DS secondhand from a friend for like $50, and it has tons of games.

I like it because it's a very immediate device no loadings, it can be closed to save battery when you have to pause the game, even for long times...What I mean is that it's the ideal console to play quick games during breaks, it doesn't always require the same dedication and amount time to get involved into the experience as on home consoles.
Than compared to the PSP I don't know. Form what I've seen and tried, it doesn't tempt as much as the DS because It doesn't seem to me quite as handy: it is bigger, it has loading times, the battery don't last very long, games on UMDs are generally more expansive and also not as portable as DS cards.
The PSPgo seems to solve all this problems but at the moment it's just too expansive and considering that the 3DS may be coming quite soon (maybe also the PSP2), it dosen't tempt me much either.
Considering games at last it seems to me that, in terms of quantity there are more quality games on PSP than on DS, but still many of those are portable verisions of PS2/PS3 games. That's not necessarily a bad thing, but choosing only one protable I prefer the DS which offers different games I couldn't play on home consoles.
With that said I don't think that the DS is immune from defects, the lack of an anlog stick can be quite annoying sometimes (Mario64 DS for example), this plus it's lower rendering power (compared to the PSP) doesn't make it the ideal platform for open-space 3D games.

Easier to play "backup copies" on :P

Well, i think that the DS have more good RPGs than the PSP. That is the reason why i bought my DS. But i like the PSP too, my friend bought one two years ago, and it's an awesome handheld too!