I think that both are essentially the same, with some minor differences.
-They can both have free videos.
-Are fully customized
*Itouch;jailbroken,
*PSP;firmware hack
-Play free games, (itouch;installous, PSP;online sources)
-Play the same format movies
-Play the same music(PSP can play WMA, while itouch cannot.)
-Essential internet speed, it just depends on your router, and connection type.
Some of the differences are that a:
-PSPs cannot have playlists, but this has been fixed with the recent updates, having its own "Itunes" experience.
-Itouch cannot play more than one format, unlike the PSP, when hacked it can play any format. This has been fixed with the release of VLC.
-PSP cannot go on Youtube. This has been fixed with an app, just go google PSPTube, or anything like that.
-Itouch cannot play flash movies online. This has been fixed with the Skyfire browser, enabling flash in an Itouch.
PSP-What this is, is that it is for hardcore gamers, evident when you cannot play games while listening to music(can also be remedied with a plugin), but it has a much more intense, widespread collection of games. These pack so much into those little discs, that they had to do data install. It has a wider selection of games, but it is mainly for hardcore gaming. It also has impressive multiplayer, while lacking the world-wide multiplaying as well as local as the DS. It is for those who are constantly into gaming, but also have some multimedia in their life. The sound and video qualities are superb, on par with an Itouch, while lacking in portability, it definitely makes up for in quality. The PSP, is also quite easy on the PC or Mac as well, all you need to do is drag and drop files over, and to access the innards, you just need to change the some settings. On the internet, it does not fare so well as the Itouch. It has a sub-par keyboard, that is bulky and stupid. There is no external keyboard, or any peripherals. There are very few custom-made sites for it, including popular sites such as Facebook, or Twitter. It uses the average mobile site. There have also been no updates for the hacked firmware. The most prominent hacker, Dark Alex, has stopped for a while now and has shown no sing of returning into play. All in all, this is a device for gaming and some multimedia added in, while it does have a large screen, and external buttons, it is also bulky and sometimes fits awkwardly into your pants/shorts.
Itouch is much more sleeker and slim, but there is very little you can actually do with it after jailbreaking it. Unlike the PSP, where all you need to enhance performance are some good plugins, you havee to manually search for the installous app, and download cydia yourself. Though this is complicated, there are massive amounts of apps that are available to you once configured. There is literally almost an app to everything. The interface for music is much sleeker, and more intuitive, as well as the movie interface. There is much customization possible, unlocked by the touchscreen. The typing interface is wonderful, as well as the autocorrect. But to customize, it is much harder than a PSP, you have to go trhough SSHing everything, and have to have constant wifi. But the results do look very good. I think that the Itouch is more for those on the go, constantly flitting from place to place, with wifi, it is much moreslimmer and portable than a PSP.
In the end, it depends on the type of person you are. If you are more of a on-the-go person, the Itouch is for you. It has a great selection of games, that you can spend time on, as well as listening to music. If you are a more local person, staying close to home, and not moving much, I advise a PSP, it has the same capabilities as an Itouch, but has a much more extensive collection of hardcore games, as well as great local online playability.