Part 3 - DS vs. PSP

Memory:

Both consoles have their own unique way of storing game data. The Nintendo DS uses game cards, which can hold 128 MB of data. In addition, game cards are re-writable, meaning that game data can be saved directly on your game, rather than on the console and/or external device. The PSP uses UMD's(Universal Memory Disc), which, in short, are discs, which are used by all current-generation consoles. UMD's can hold nearly 1.8 GB of data(wow!), meaning the PSP is more ideal for developers like Square Enix, who is very popular for many RPG's, such as Final Fantasy, Chrono Trigger, and Kingdom Hearts. Unfortunately, UMD's aren't re-writable, meaning game data has to be saved externally on a memory card. Also, the PSP can download games, which can be saved on the memory card. The DS cannot download games because it doesn't have enough memory on the console itself, and has no external memory input.
The DSi(this is not a typo. The I in DSi is lower-cased.), which is an advanced model of the DS, CAN download games, due to an external port for SD cards. But, since the DSi is often considered an entirely different system, we're going to pretend it doesn't exist.

Winner: PSP

Music Quality:

Both consoles have the ability to output MP3 files in games, meaning both consoles CAN(and I use that term freely) output fully orchestrated music. In addition, the PSP has a function which allows you to listen to MP3 files. I'm gonna let that statement finish itself.
Another thing to note is that the DS, due to it's 128 MB limit, couldn't have as many songs in the games soundtracks, meaning the PSP was more ideal for people who wanted a more cinematic sense of gameplay. I'm tipping my hat to the PSP this time, but, otherwise, this one was a pretty close call.

Winner: PSP

Backwards (and Cross) Compatibility:

Backwards Compatibility means a system can play games that were released on previous-generation consoles. The DS has an extra game cartridge slot for Game Boy Advance games, which is awesome. I still use my DS to play the Mega Man Battle Network hexalogy(hexalogy means 6 pieces of media related to each other. e.g. Star Wars 1-6) to this day. Since the PSP is the first in the line of Sony handhelds, there is no previous console for it to emulate.
You CAN download certain Playstation One games off of the PSN store, such as Tomb Raider, Metal Gear Solid, Tekken 2, the Spyro Trilogy, etc.
The DS wins because it can play EVERY Game Boy Advance game, as opposed to the PSP, which can only play select PS1 games. No Tetris Plus here, folks. =(

Winner: DS

Continued in part 4