What's the differences between "due to" and "but for"?

"Due to" assigns a cause.
Example: The high number of accidents yesterday was due to icy roads.

A similar phrase is "owing to". Both phrases have the sense of a debt.

"But for" precedes a necessary condition.
Example: But for the long hair, that man looks like my brother.

This means that 'That man' looks like brother, except that my brother has shorter hair. A true description of my brother would necessarily include 'short haired'. But this man has long hair. Therefore he is not my brother.