Be sure to check Dreamkatcha's list for games that have had remakes on those consoles:
The DS scene has been much more active regarding the number of homebrew games.
To add to Lobotomika's post, with the DS you more or less won't emulate properly above the 8-bit console generation. For instance, the MegaDrive emulation is pretty good but it fails on the issue of the resolution. The screen will either be cropped or scaled (making text, life bars etc hard to read).
Ofc, if you want some Mario Kart derivatives, you can have your fair share on PSP too
As for regular titles, it will depend on what you like and how you think you will use your handheld.
Many PSP titles are ports/conversions/sequels/prequels of titles you can find on the home consoles (PS2/Xbox360).
If you enjoy those and think it would be nice to play them on the go, then the PSP is for you.
But then, you might think it is better to play such games at home.
DS has a more versatile library I believe.
Owning both, I use:
- PSP for emulation (even GBA, despite having a DS lite with an EZ3-in-1 card which allows me to run GBA ROMs as I enjoy being able to save state in some games). And a few regular titles, mostly Koei's Warriors series that I enjoy very much when travelling.
- DS for homebrews (including remakes), puzzle games (either Nikoli-style or interactive style such as Professor Layton), Shogi games (PSP has some but I find it more agreeable to play on the touch screen), some RPGs (Radiant Historia might be the best RPG I have ever played), Yu-Gi-Oh Nightmare's Troubadour (yes, it is for kids, but it is really enjoyable. And I am normally a fan of TCGs), the Castlevania games, some strategy games (Panzer Tactics)