Wheat bread used to be considered the best pick for healthier sandwiches. But with the arrival offortified breads, health experts agree that even the white variety can provide near-equal health benefits of wheat. Here's why.
When comparing whole wheat directly against white bread, whole wheat appears to be better. The degree of which it is better, however, is uncertain. Some sources claim little to no difference, some epidemiological studies claim much more.

Increasing dietary fiber seems to be implicated in many of the benefits, and can be achieved through increasing plant intake in general (fruits and vegetables included).
So if you prefer white bread, you no longer have to worry about sacrificing nutritional value because you don't want bits of wheat stuck in your teeth.