I own both systems so I will give you advice from my experience.
- DS Lite -
Games: The first game I purchased for it was Meteroid. This game gave me a very false sense of what the DS could do. "I was like, hey, this is passable 3-D, I've been wrong about the DS." However my hopes were quickly shattered when I played nearly every other game for the DS. The graphics quality of the DS on average is something like a very ugly and under polygon'd N64 game. I was never a fan of the N64 (even though it had carts, I tried to love it, but couldn't) but if you are a big fan of the N64 and it's look has some nostalgia value for you, this might be a plus. The color on the games however is usually fine, it's the just the overall graphic quality that will make you want to

. The DS is capable of playing FMV in the games, but it is usually less than 10 minutes per game. The quality is better than Youtube, but not as good as FMV on the PC-FX. I am left handed as well and many of the games have no left handed mode making most of the gameplay awkward at best. I often am forced to use a fingernail instead of the stylus because of this. The budgets on quite a few DS games are very small since they are aimed at younger people (IE crap). The system does have great first party games though. A few names stand out like Meteriod, Mario, and even good old Sonic. The games are usually 20-44.99.
Backwards Compatability: The DS also plays GBA games. However these games stick out of the unit when you play and you have to find a place to put the bottom slot cover while you play one of these games. The unit comes with no DS slot cover, so if you only wanted to play GBA games for say a week, there would be no way to cover the other slot. If you do not have an afterburner in your GBA you will be finally able to see games like Castlvania thanks to the backlight!
Construction Quality: Both the DS and PSP have problems with occasional dead pixels. The DS controls however are make for smaller hands than the PSP. The trigger buttons feel very cheap and sometimes get stuck in a slightly diagonal, pressed in mode. You can pry them back, but it might really mess up your gameplay in many games since you have to be very careful and press these buttons very lightly. Many, many DS users crack the back hinge through casual use and play, treat this thing like it is made of eggshell.
Homebrew/Web/Other Extras: If you want to run homebrew on the DS you are going to have to invest in some kind of cart which takes a mini/micro SD card (usually). You will have to buy the memory separately. To browse the web also requires you buy an expansion card with more (slow) system ram. This ram can be used for some homebrew and to web browse. You must also purchase the web browser cart and probably a wifi connector since the DS doesn't really come with wifi. The DS can link with another DS however to play multiplayer games. The homebrew on the DS is mostly "cute" rather than "breath taking".
Screen(s): The screens on the DS are a step up from the screen used on the GBA (I do not own a SP, so I have no idea how it compares). They blur in some games, but not badly. The color seems crisp, you HAVE to buy a screen protector for the lower screen though or over time it will become utterly scratched up from gameplay. If you have not used a DS before, I should note only the bottom screen is a touch screen.
Battery/Charging: The battery in the DS Lite will run my unit on the lowest brightness, 75% volume for about 5-7 hours. The charging time is probably 2 hours on the Nintendo adaptor. The plug on the adaptor however is pretty funky. Make sure the DS will not get yanked or knocked off something while this is plugged into it. The connector looks easy to screw up.
- PSP Slim -
Games: Most games on the PSP have a playstation sized budget for their development and because of this the games often look somewhere in between a PS1 and a PS2 game. Some games look nearly as good as a PS2 games though which is pretty impressive. The PSP also tends to have wonderful music quality since most games are the size of a CD or larger. There are many big name console games present in PSP form like Silent Hill, Metal Gear, Wipeout, Ridge Racer and even
Super Puzzle Fighter II: Turbo. 99% of the games have clear, crisp FMV. Built in wifi allows you to play multiplayer games from different rooms in the same house. Games cost between 20-40 dollars. It should be noted if a game is stepped on, the shell might break, thankfully you can order a new one online which is more durable should this happen.
Backwards Compatability: The PSP is capable of playing PS1 games off the memory stick. While it can play 95% of the PS1 games fine, Sony would like to restrict you to only playing a small selection of games, which grows over time. I guess this way the average consumer is more impressed or something. See homebrew paragraph for more PS1 info. The PS1 games play fine on the PSP for the most part, some like FF8 still crash at times. Everything is recreated faithfully and the graphics are improved, much like playing a PS1 game on the PS2. Load times are small thanks to the fast speed of the memory stick.
Construction Quality: The PSP Slim is lighter than the PSP fat for a reason, much of the metal structures which were in the PSP fat to strengthen it have been removed, it still feels more solid than my DS however. There are no problems present in the buttons or d-pad or triggers in the slim and the controls will comfortably fit anyone used to the PS1-3 line of controllers. Be careful of the UMD door since it is probably the weakest part of the system. The power switch on the PSP slim is of better quality than the switch on the PSP fat, this was always one of the weaknesses in the fat as it would fail with years of use.
Homebrew/Web/Other Extras: The PSP can browse the web from the start. Nothing is needed. You can also get RSS feeds, streaming audio broadcasts, play mp4 movies, mp3 audio, wma (if you felt the need) and UMD movies out of the box. The Slim also has a TV out, however you must buy a cable and you must have a progressive HD TV to play games on the out, otherwise you can only display XMB functions like the mp3 player, the movie player, or the web browser. If you install a custom OS to the PSP however you can play nearly any PS1 game you own, you can play popular arcade emulators like CPS 1 and 2, Neo Geo MVS, and older things like the GBA or SNES. With a custom OS you through homebrew programs can play tracker files, wav files, oog (ogg? I forget), Divx and other formats. I personally love playing Sewer Shark on my PSP, it gets really funny reactions from people in public.
Screen: Some PSPs like the DS come with occasional dead pixels. You will want to buy a screen protector so it does not become scratched. The screen however is large and bright. It is very responsive and blur you see in some games is usually (if not always) from the game itself, not the screen. It is possible to adjust things and see this in Ridge Racer for example. The screen is actually one of the best things about the PSP.
Battery/Charging: The Slim comes with a 1200 battery, this battery is fine but you really want to replace it with a 2200 fat battery (if the stock is new it comes with a slim conversion cover, otherwise you can buy one for under 3 dollars with shipping from dealextreme). Then you get easily 6 hours or more of play out of the PSP, even with high brightness (level 3) and 100% volume. Charging the 2200 battery takes about 2 hours, I don't know for the stock one. It has been unused for some time. If you have a custom OS, you can also adjust the clock speed of the PSP which is 100% safe as the cpu and bus are designed to be re-clocked on the fly by the software. This will either add to or shorten battery life. It is very useful though on a road trip if you want to play MP3s because you can clock it all the way down to 75MHz and still play them.
In the end I go with the PSP because it is so much more powerful, but with a Sony OS, you are kinda wasting most of the potential of the PSP. Each unit completely stock, it's kinda a tossup, but I still think the games on the PSP are by far the best.