My response to anyone comparing the DS to the PSP (this is not neccesarily directed at you, Zera... more at a general audience, though it should still more or less answer the question): I think it's mostly a matter of preference. The main concern when choosing one over the other is games. The PSP's library of games is quite different from that of the DS, for one thing. There are few DS games which interest me, so I got a PSP over a DS. If you don't like DS games, you shouldn't buy a DS. If you don't like PSP games, you shouldn't buy a PSP. Why buy a game system that doesn't have games you like? That's the main concern, but if that doesn't lead to a decision, the other factors become more important.
There is certain functionality which the PSP posseses and the DS does not (though some of it can be added to the DS), but this functionality may not matter to some people...
Do you want a portable that plays music and video more or less out of the box? If not, you won't care that the PSP can do this and the DS can't. I do, but again, that's just personal preference. Do you care if you can browse the web on it? The PSP can do this out of the box, but the browser is clunky and not all that great. The DS doesn't come with a browser, but you can put Opera on it, which blows the PSP browser out of the water. The stylus is a much better tool for surfing the web, too. I rarely use the PSP browser, so it's not a big deal for me. Don't really do multi-player either, but the PSP seems to possess more functionality in that regard. Want something that can interface well with other hardware? The PSP does this quite well. It can interface, to varying degrees, with other Sony hardware, with an Xbox 360, and with computers (works with both Windows and Mac OS X, maybe others but I have no experience there)... just about anything with a USB port. To my knowledge the DS does not do so well in this department, but again, you have to ask yourself if you actually give a rat's ass about this particular aspect of the handheld.
How about battery life? Pitiful on the PSP, because the screen is bigger and brighter (you can dim it, but this is a poor comromise), the system is more powerful and thus needs more juice, and spinning a UMD is far less efficient than reading a cartridge. The DS is very efficient and lasts a long time on a full charge. I don't really mind. You might. Note that the PSP can be placed into sleep mode at any time, reducing power consumption by 99%. Battery life varies wildly... play online with the screen at full brightness and it will go in a few hours... use it as an MP3 player and it'll last all day and night.
How about durability? The DS is very resilient and doesn't really have moving parts. The PSP is relatively fragile in regards to physical shock, and does have some moving parts. The screen is vulnerable to scratches (protect your investment with adhesive screen covers.. thin, transparent plastic that covers the screen and will protect it from damage... replace as needed... these should not be considered optional). Dropping a PSP is a big deal, dropping a DS is not. I never use my PSP without the wrist strap. Ever. Doesn't matter if I'm sitting on the couch. I keep it in its pouch when not in use.
How about the games? UMDs are bigger than cartridges. The can get scratched if not in a case or some kind of sleeve. Memory? You'll likely want to buy some for a PSP. Up to you how much. I personally like the fact I can choose how much memory I want, can store different things on different sticks (I only have 2... but if I wanted I could put savegames on one, and media on the other). I like the fact the memory sticks can be used with other stuff if need be. Does this appeal to you, or is it just a waste of time or money? In the latter case, it's not an issue with the DS.
Do you sweat much while gaming? My hands tend to sweat slightly. If you get a PSP, be aware that they smudge like you wouldn't believe. I have little personal experience with the DS Lite (it's too small for my gorilla hands!) aside from time spent working at a game store, but I don't think it can compete with the PSP in terms of smudge-magnetism. The DS isn't even on the scale, there. Thankfully, moving the PSP's power switch into the 'HOLD' position will effectively disable all the buttons on the PSP, allowing you to wipe off those smudges whenever you please. Still irritating, though. If you're anal-retentive, you may find the amount of time it can keep you occupied is slightly increased due to smudge removal breaks.
In short, the games are quite different, the PSP is far better for media but far worse for web-surfing. It uses more power and needs to be treated with a little bit of care (nothing unreasonable, but considerably more than a DS... the DS is built like a tank). It is prone to harmless but irritating smudges which you can wipe off in a few seconds, if they bother you it all. It works well with other hardware, and is much more versatile in regards to memory.
As for games... the PSP is interesting in that most of its games, in my book, fall into one of three categories; 'garbage', 'meh' or 'wow, this is a good game'. Some of the 'meh' games are not actually bad, they just have narrow appeal. You'll have to brave the dangers of that category yourself! As for the third category... the PSP... again, just IMO... makes up for its otherwise underwhelming library with these gems (some of these metaphorical gems are not perfectly cut or polished, but they're still gems).
Killzone: Liberation is astounding... the boss fights are very, very hard, but I reccomend it in spite of that. It's hard to compare it to anything... it's a pseudo-isometric shooter, built for the ground up to be played on the PSP.
Lumines is a true masterpiece. At the risk of being executed for heresy, I will say now that I prefer it to Tetris.
Syphon Filter: Dark Mirror shattered my perceptions of what is possible on a portable system. It's very rare that I can find nothing bad to say about a game... but this is one of those occasions. If you like shooters, you will probably like this.
GTA: Vice City Stories is one of those unpolished gems. The controls are a bit troublesome at times, and sometimes it almost seems too much for the PSP (occasional audio hiccups and trouble keeping up a times in terms of graphics), but it really is GTA... if you like GTA, you will almost certainly like this.
Hm, Wipeout Pure is great if you like racing games... actually, it's pretty good even if you generally don't. It's the only game in its genre which I actually enjoy. You can find very cheap used copies, so it's worth a shot.
Those are the ones that I am quickest to praise and to reccomend, but there are other games on the PSP which have also recieved stellar reviews. These games, in my opinion, more than make up for all the dog crap in the PSP library. I would reccomend that you give them a try before writing off your PSP. If you don't like them, PSP probably isn't for you, but they are held in high regard for good reason.
R(Tekken