Mini is when you want to extend the TiVo experience to a TV elsewhere in your house, it has a monthly fee associated with it or you can get lifetime
Stream is when you want to extend the TiVo experience to a tablet or iPhone elsewhere in your house, there are no monthly fees associated with this device
The Mini requires a 4 tuner TiVo Premiere to configure and permanently takes one tuner away from the host DVR to watch Live TV (This will change soon), of course you can configure it to not support Live TV at all and only watch previously recorded shows.
The Stream can watch Live TV, sort of. When you specify that you want to watch Live TV, it tells the host TV to record that show then plays it back. The downside of this is that it will leave the recording on the Host TiVo when it's done, and it can get a bit messy. The Stream also supports downloading the TV show so it can be viewed offline (Say on a plane, or elsewhere outside of your home), but this this subject to the whim of your local Cable provider and how aggressive they are with setting the copyright flag. Cox and Time Warner are notoriously bad for this, Comcast is better and FiOS is best.
I current have a Stream and love it, I'm waiting on the new TiVo's to drop this fall to upgrade my oldie 2 tuner Premiere to a 6 tuner beast and will replace my secondary Premiere with a Mini.