The Avenger and Oddessy are on the opposite ends of the spectrum.
The Oddy is a tank. Big, honking tank. Unlike the usual, the C-Store versions are better as opposed to the Star Oddy, which you can get via Fleet req. Big hull strength, 1.15 shield mod, universial LtCmnd station. Also all three of the Oddy Consoles are pretty good. However it turns like a tortoise in molasses. It is a beam boat; mounting cannons on it will do you little good.
The Avenger is faster, more aggressive and has a 5/3 weapons set up as opposed to 4/4. While it plays more like an escort (Faster, most of the weapons in the front) it has a lower turn rate compared to escorts of the same tier. Sure, with strategic maneuvering you can bump it up a bit, but I prefer Weapons Efficiency. With a mix of Dual Beams and Arrays, you can make it a great, speedy, Beam Boat. However compared to other cruisers it's not as durable, and has a lower crew count (And thus, hull regen).