The difference can be compared to the difference between regular DSL and Broadband connection speeds.
Edge is an old architecture that is slowly being phased out, and 3G will begin to be replaced in 2010.
But to directly answer your question the main differences are as follows.
On 3G you can talk and browse the web/ use data at the same time -you can only talk and text on Edge.
3G speeds are much faster than edge. I recently tested my 3G speeds and was getting an average of 1400 kbps compared to an average of 120 kbps of edge download speeds.
To answer your other question, the iPhone 3G and new 3Gs is equipped to utilize both. the default is that 3G is on and will switch to edge when 3G is unavailable. 3G uses more battery, so if you are mostly making calls, it's better to turn 3G off and use edge. If you use more data than voice, leave 3G on.