It depends on what you look at.
Epidemiological studies are surveys. They can be done all at once (cohort) or done over time (longitudinal). They aim to find correlations between multiple variables (things looked at).
Many epidemiological studies suggest that there is a benefit associated with whole grain intake typically associated with dietary fiber. These large scale studies tend to be confounded with lifestyle however, and are subject to varying ways to define and measure the concept of 'whole grain intake'. One study, using FDA approved definitions of whole grain, found no consistent association between whole grain intake and cardiovascular disease or diabetes prevention.
There does seem to be a relation between whole grains and health, but whether these are due to the whole grains themselves or habits associated with whole grain intake is still up for debate.
Intervention research tends to be the type most people think about, looking at two groups, introducing a change to one group, and seeing the changes that occur in the latter group (experimental) relative to the unchanged group (control).
The results of intervention research suggests no benefit to cardiovascular disease prevention
Relative to refined wheat products, whole grain products (rye is highly studied) tend to increase satiety (the feeling of fullness) which may be due to higher fiber content. It should be noted that Rye bread seems better at this, and non-rye whole wheat bread is more unreliable in its effects on satiety. This increased sensation of fullness may be a reason behind the association of whole grain intake and lower body weight in epidemiological studies.
The glycemic load of white bread is within 10% of that of whole wheat bread. The difference is explained when factoring in that white bread usually has 2-4 grams of fiber for two slices, whereas whole wheat tends to have 4-8 grams.
Nutritionally, with the advent of fortified white bread, the two are very similar in micronutrient profile. So on a nutritional level, there isn't too much of a difference between one serving of white grain product and a single serving of one whole wheat product.
Whole wheat bread tends to have a higher fiber content, and more micronutrients than standard white bread. Fortified white bread can close the gap between the two with nutrient fortification, but fiber tends to still be different.
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).