A contractor is a specialist in his or her field: electrical contractor, plumbing contractor, construction contractor, etc. They often oversee a team that may consist of subcontractors who handle specialized jobs within the larger project. They are expected to be knowledgeable about permits and other such technicalities.
Handymen specialize in odd jobs. It could be replacing a broken cabinet hinge, stopping a leaky faucet or sanding down a gouge in your wood floor. The handyman is a trained generalist, and it was probably a handyman who coined the phrase, "No job too small."
To determine if you can use a handyman instead of a fully-licensed contractor, ask yourself the following:
Could I do it myself if I only had the time?
Could I do it myself if I only had the knowledge or ability?