WD Has a very nice Chart, spelling out the differences between the three
Here's the Link
But just to give a brief over view,
The Mini is the most basic model, with a single USB port and has 480P Component and Composite out, i.e. no HDMI.
The
WD TV HD is a step up, supporting 1080P out over both HDMI and Composite. it has Two USB ports, allowing you to plug in two seprate drives. I have this model, and have to say it works pretty well. It is however the Older of the three models.
The
WD TV Live is the only one of the three which has a network connection, via Ethernet or Wifi (Using a separate Adapter). It has all three connections, giving maximum flexibility. It can connect to you tube, Pandora, flickr, etc. Best of all it can access shared files on your home network.
On which one to get, I'll definitely suggest paying the few more dollars to get the live if you are planning to go for TV HD, as the additional features are more then worth the additional price. If on the other hand you just need a device which can support SD Playback from your hard drive and don't want HDMI then the Mini is a nice solution.
I got the TV, before the Live was announced, and have to say its serving nicely as a box to play my downloaded and ripped content of a Hard drive.