Ok I'll give you a "in a nutshell" review, I own a PSP2000 and a NDS Lite, I love both, but here are the difference.
Graphics: PSP wins
Hardware: PSP wins
Sounds: again PSP wins
In reality, you should not care about hardware, the hardware is only as good as its games, if there are no games to play, I can give you a system with 10 cores and graphics that tops today enthusiast PC graphics card, which in the end will do absolutely nothing if you don't have any games you want to play on it.
Do remember that they are of different breeds, it's like trying to compare an apple with an orange.
In my humble opinion (IMHO), PSP is great for sports games, a few action games (FPS, RTS, etc.), but the NDS has some great RPG / Adventure games, some will say that NDS will be better for girls since it has a touch screen that you can poke at with a pen, also NDS has a lot of cute'sie games like cooking mama, pet chihuahua, etc..
Really it boils down to your flavor of games, if you like games like Monster Hunter, Final Fantasy Crisis Core, FIFA, Dynasty Warriors, Metal Gear, then go PSP..
If you like games like Castlevania, Final Fantasy (III,IV,1 & 2 which are the classics), Zelda, Pokemon, there are too many RPG's to list, the touch screen does add a different style of game play which is unique compared to PSP..
Now regarding homebrew software, both supports it with the proper mod for PSP, R4DS or similar for the NDS, I wont dabble too much in this but you can technically make both a portable media/MP3 player, given the PSP will have a advantage in this area due to the hardware specs.
As for free games, I don't know of any specifically for the NDS, PSP you can join the Playstation network which will allow you to download demos or trail runs of some of its latest games, mind you that if you mod your PSP, you may or may not be able to join the Playstation Network without updating to its latest firmware, which in turn will overwrite your PSP mod. (or are you referring to "free" = pirated? use Google)
An NDS lite (non local distributor) without an R4 cost around $900, PSP cost more, the lastest models are the 3000 and the PSP go, don't think they have managed to mod the GO yet, but the 3000 can only be modded via a emulator, 1000 (the big fat brick) and 2000 series can be flashed via firmware and hence just that, finding it used actually cost almost as much as buying a 3000 new.
I don't recall why I bought a NDS Lite that was not warrantied by a local distributor (my old one broke after dropping it like 10 times over the period of 2 years so recently replaced it), mine came from Korea, think I was told that the warranty/repair process was a royal pain in the rear, the offical local distributed one at the time was $1200 compared to $900 which is what I paid for mine.
I bought my PSP too long ago so I don't remember how much I paid for it.
Both the PSP and NDS build quality is pretty tough, sure if you treat it like your step child and drop it on it's head a few times, it might be a couple sandwiches short of a picnic afterward, you get the idea.
Hope this helps