In the United States, very little. For example, we speak of BOTH "hamburger rolls" and "hamburger buns." I think "buns" is more common, but I'd have to go to the supermarket and check the bread section to be absolutely sure. This is also an area where there are strong local differences. Even if you didn't use the local choice, everyone would understand you.

Hmm...hadn't thought of the distinction...good question. :) Normally, we use "bun" when it's sliced in the middle and we're putting meat in between, like a hotdog or hamburger bun. But we say "roll" when it's something a solid thing that we would break apart and butter during a meal, like a dinner roll.

Terminology varies from one region to another, but this is a British perspective: A roll is like a miniature loaf of bread, and is probably oval or torpedo-shaped. A bun is round. Hamburgers come in buns. A bun can also be sweet - for example if it has dried fruit and spices in it.