The difference is this: In Alternate Shot, the players alternate hitting tee shots. In Odds and Evens, one player hits the tee shot on even holes, the other hits the tee shots on odd holes.
But wait, you say, isn't that the same thing? Almost. In traditional alternate shot, the players on a team get to choose who tees off on No. 1, and alternate from there. In Odds and Evens, the player who tees off on No. 1 is chosen by some random method.
It's a very, very small difference that really just takes one decision out of the hands of the players.

