Theme header is as complex part as whole separate layout, so it has its own groups of various settings. You can find all of them in Appearance → Customize → Header Settings.
As navigation it also has tools to search, open different parts of layout like Hidden sidebar or cart from WooCommerce.
You have access to 2 main types of theme header: vertical & horizontal. Depending on this choice you will get access to different options in other theme settings as they have different possibilities.
It is hard to list them all differences, so if any theme option works only with one type of header we will inform about it in its description.
Each header type also has different variants so it adds even more possibilities for you.