What is the difference between Opera Mini and Opera mobile? What features does Opera Mobile have the Opera Mini doesn't? Are there any other differences such as speed?
Opera Mobile is a "full" browser performing all rendering activities on your Android device.
Opera Mini passes your requests through Opera's servers to render/scale/"tweak" for your device.

Opera Mini tends to be faster.
Opera Mobile tends to hit your device's resources harder.

Security folk cringe about Opera Mini and your traffic going through 3rd party (yet Opera) servers. Think about that before doing your online banking with Mini... 

Read more in the comparison chart here: http://www.opera.com/mobile/features/