Opera Mini is a proxy browser. Any website you with to view on it first passes through the Opera servers located in various locations around the world. The servers strip out overheads, compress and pre-format the webpage, effectively an image of the webpage is then sent to the Opera Mini running on your telephone. The fact that on Opera Mini you are not looking at the full and directly transmitted, live webpage limits some of the user interaction you can have with that webpage. If you are paying for data by the MB or have a low data limit, Opera Mini is definitely the best option and will save you plenty of money. I also find Opera Mini usefull if I find myself on a slow GPRS connection, it kind of spees things up because the amount of data you are waiting to load in is much smaller than using a standard mobile browser.
Opera Mobile is an internet browser in the true sense. It makes a connection with the server of the website you want to view with no other processing inbetween. Opera Mobile then renders the webpage on your mobile device. Because Opera Mobile is making a direct connection and is showing you the "live" webpage it can allow for greater interaction with the webpage and the viewing of video etc. Opera Mobile does have the option of using Opera Turbo. Opera Turbo compresses webpages in a similar manner to Opera Mini for reduced data usuage and therefore reduced cost if you pay per MB.