What is the difference between a branch and a level of government?

A Branch of government is a governing entity concerned with a specific category of government service. Such as: National Park Service, US Dept. of Education, State Agriculture Dept., State parks Dept., Municipal Clerk, County Engineer ... etc.

A Level of government refers to both the magnitude of a governing entity's jurisdictional authority as well as the extent of it's geopolitical territory. Such as: City, County, State, Federal.