What is the purpose behind building a covered bridge instead of just a bridge?

To protect the trusses from the environment

Old bridges were made of wood and when they get wet, they rot. The cover (roof) protects the wood like a roof protects your house.

Yeah I believe it protects the structural components of the bridge

The roof sheds the elements, like rain, snow, and ice preserving the structure.

I'd guess it's related to protection of the portion of the bridge that people would walk or drive on.