I'm a lucky enough to own both a Sony PSP and Apple's new 5th generation iPod (the video ones). My iPod is a 60 GB model in black. I'm not posing side-by-side pictures along with this post because you can find gazillions online.
The PSP is an awesome portable video game device. It's far too large and cumbersome to be worth anything as a music player. The iPod is an awesome music player, but is too small and too weak to be a serious contender as a gaming platform. So why do people keep pitting the two against each other?
Because the demographics for these two devices overlap in a significant way and many people will be unwilling or unable to buy both. So these devices are competing for sales to some degree.
I've watched significant amounts of video on both devices at this point. I've watched UMD discs on the PSP as well as MPEG4 video saved to a memory stick. I've also taken in TV content from the iTunes Music Store as well as *gasp* DVDs I've ripped to watch on the iPod. These are, of course, DVDs I purchased. The DMCA still says I'm breaking the law though.
The shocker here for me is that despite the smaller screen, the iPod is easier and more enjoyable to watch video content on. With the PSP, I have to carry around a bunch of UMD disks and/or memory sticks if I want to be able to match my mood when I start to watch something. The PSP is too large to comfortably hold in one hand. Due to its shape, it's also does not do well when placed on a flat surface in front of you. So, you have to hold it with to hands, prop it up on your leg or find some other arrangement that allows you to view the screen from the right angle.
The iPod just sits in the palm of your hand. All your video is sitting on the spacious hard drive so you can have an extensive library at your disposal with no space penalty. All we need is feature-length movie downloads from the iTMS or a legal, fast way to rip DVDs and the iPod will achieve consumer electronic nirvana.