A banner is a rectangular advertisement place either above below or on the side of a website.  It usually is a link that takes you directly to the advertisments website.

A header is the top rectangular shape that runs accross the top of the web page.  The purpose of the header is to promote your comapnies brand and make it instantly reconizable to your audiance.

The difference between the two is that in a bander it is promoting a different product and is providing you a direct link to that products website.  In a header it is attempting to catch the audiances eye and get them to continue to look at there website.