what are live and dead centres?
To put it simply a dead center is just that - dead. It has no moving parts. It is really nothing more than a metal shaft with a point.
A live center is similar, but the shaft has a bearing that allows it to turn. For example, in my lathe, the tail stock has a Morse taper, in which I put a live center. The tenon fits in the Morse taper, and it has a bearing assembly with a conical point on the other end that is free to spin.
(source: Z Live Center)
What is the functional differences between a live/dead centers?
The live centers are great in that they don't have really any friction and allow your piece to spin freely.
Dead centers are more often found in treadle or pole lathes, which are often built to be "period correct." The use of a modern live center would be anachronistic in this case. A turner would use the dead center with some grease or oil on the point to keep it from squeaking in use. I've also heard that ear wax or the skin oil on the side of one's noise was historically used, but don't quote me on that.
Would there be a practical reason to choose one of the other?
If you're not worried about being "period correct," the live center is really the way to go. I think most modern lathes ship with one standard, anyway.