I own an ATV 1 and an ATV 2, here's my thoughts on all the ATV models.
Apple TV 1 Advantages:
- Hard drive to sync content to, so you can just sync your content to ATV or take your Apple TV to a friends house.
- Supports component video, and composite (either with a Firecore-style jailbreak, or by plugging a shorted-our HDMI adapter in at exactly the right moment while the ATV switches resolution.
Apple TV 1 Disadvantages:
- No access to Hulu Plus or Netflix.
- Built to sync content from a Mac. Doesn't sync music and video from iTunes Match.
- Hasn't been updated in a very long time, never will be in the future.
- Very unreliable power-wise. If you put it to sleep, it'll keep waking itself up.
- Runs very hot and uses a lot more power than the ATV 2 and ATV 3.
- Doesn't do AirPlay (video), but can do AirTunes (audio).
Apple TV 2 Advantages
- Syncs and streams purchased content from iTunes, no need to sync with a Mac.
- The newest Apple TV that can be jailbroken (seas0npass and ATV Flash), adding extra features like XBMC and the ability to watch videos on a network share.
Apple TV 2 Disadvantages
- Very expensive to acquire, since they're not made any more and it can be jailbroken, unlike the Apple TV 3. Originally sold for $99, but expect to pay $150 - $200 for one on eBay.
- Only outputs video in 720p, not 1080p.
Apple TV 3 Advantages
- Syncs and streams purchased content from iTunes, no need to sync with a Mac.
- Can output video in 1080p.
Apple TV 3 Disadvantages
- Unlike the ATV 2, it can't be jailbroken. They've been trying for a long time and
still haven't gotten anywhere.
Edit: Both the Apple TV 2 and Apple TV 3 can do AirPlay.