A civil engineer is usually consulted to advise on the structural integrity of the plans for a build. A general contractor is usually responsible for overseeing the construction and making plans for the timing of the construction. They also usually make estimations for the build based on the design.
A civil engineer will usually only play a small part in a build, by helping the architect to finalise the plans and make sure they are safe and structurally sound. The general contractor plays a much bigger role in the project and will usually be present throughout the duration of the project.
For example, I employed a general contractor in Dartmouth, MA to build a sunroom on my house and they oversaw the whole project from start to finish. The general contracting company will likely liaise with a civil engineer if necessary.