Apple - What can a 1st gen Apple TV do than a new Apple TV cannot?
The only super major differences the 1st Gen Apple TV's has from the new ones is the ability to store Movies, Music, Content, etc because they have a physical hard drive in them for doing those things. The other things is that they can connect to TV's through Red, Blue, Green Component Cables. They also do not support Airplay with out a hack or a software fix. Also, the newest Apple TV, the 3rd Gen, allows for 1080p HD playback, where the older ones only support 720p HD playback.
There are also a few other minor differences in size, power consumption, software options/OS and so on.
The new Apple TV's do not have hard drives or Component Cable outputs and support Airplay out of the box. Airplay is probably the biggest feature that makes people who don't want to hack to jump ship to the newer box. It is pretty cool. The 1080 HD playback is nice, but depending on your TV, may or may not be noticeable/important.
Some people like the older ones also because they can be hacked and used as linux computers (link) with actual storage and since they can also store things on them, you can take it with you and have videos/music on it and not require internet for streaming your content.
Though the biggest reason is most likely the hack-ability of them, they are tiny computers with a decent amount of power. And since they aren't sold any more and are still sought after, that would explain their above average price.
The new Apple TV's require internet of some kind or at least a local network for stream video and movies and data from a computer or the internet.
They are also significantly smaller too, the older Apple TVs are much larger.
You can see all the differences on their wiki page, it lays out the differences in features between the devices and even talks about most of the hacks that are available. Very informative.
I hope that this answer helps you out.
de_an777 wrote a pretty solid answer. I'll add that because the underlying CPU is an Intel Pentium-M, the first generation Apple TVs can be hacked to run OS X. The later generation ones are ARM based and are thus restricted to Linux distributions that are ARM-compatible.
Here is the big reason: The first gen ATV has a wifi card which can be removed in favor of a $40 Broadcom HD Decoder card. Then, with a customized Linux distribution, you can run XBMC on the device and stream 1080 content from a NAS or another computer.
Since there is no jailbreak currently available for the third gen ATV, this is an easy an inexpensive way to watch HD video on one's HDTV, complete with an IR remote, and it's a full linux install as well.