As I understand it annealing restores the brass' elasticity which will have an affect on neck tension once it has been put through a sizing die.
Actually, it's just the opposite. Work hardened brass, the kind you get after one or more cycles of resizing and firing, is more elastic, not less. Most metals, including brass, will bend a certain amount under stress and then return to the original shape. This is elastic deformation. Things we call "springy" exhibit elastic deformation. Rubber and most metals exhibit elastic deformation if they aren't bent too far. If sufficient stress is imposed on the component, it will deform and NOT return to the original shape. This is plastic deformation and it generally happens when a metal is pushed beyond it's elastic limit. A paper clip will be a good test subject if you're prone to experimentation.
Plastic deformation is what is supposed to happen when we resize. If the brass didn't deform permanently, you couldn't call it "resizing". If cases are sufficiently work hardened, they will be too elastic and they will tend to spring back toward the original shape. Not completely of course,but at least to some extent. With the force of our presses, even work hardened brass will undergo plastic deformation, but the result is not identical for work hardened brass when compared to annealed brass.
When you anneal brass, it becomes softer and more ductile (less elastic) so that when you resize, the brass is reshaped with less "spring back" than it would exhibit had it not been annealed. For this reason, and contrary to some claims in this thread, annealing does indeed have an effect on neck tension.
Some people, including me, anneal (with a very precise automatic machine) every time; not so much in an attempt to achieve a particular neck tension but more in an attempt to equalize the neck tension in every case in that all elusive search for consistency which is generally regarded as the holy grail of reloading.
Work hardened brass has a higher modulus of elasticity, which is a measure of how "springy" it is. It is also stronger, less ductile, and as far as we're concerned, prone to cracking if deformed one too many times.
Bottom line: Annealing will make the cases "less springy", less prone to crack, and (we hope) help produce more consistent neck tension.