While they do overlap, grammar checkers are software programs that scan your writing for mechanical errors (e.g., grammar errors; punctuation; spelling; and basic clarity). Grammar checkers will scan your writing and identify these types of errors within a matter of seconds.
Proofreading is the human review that occurs after your writing has been checked with a grammar checker and is where you (or another person) closely read the document for the things that software cannot identify; this includes checking for tone changes/variations, awkwardly structured sentences, factual errors in the content, formatting inconsistencies or errors; and ensuring that your writing means what you wanted to convey.
Combining the use of both grammar checkers and proofreaders is usually the best way to prepare documents. Using a grammar checker first allows you to eliminate the obvious mechanical problems in your writing so that a proofreader (human) can identify problems usually overlooked by software. If you would like a more detailed explanation of what a good grammar checker contains, this FAQ covers the components of a good grammar checker.