Traditionally, proofreading has always been carried out prior to final printing. Before the digital age, printing required a great deal of time (and therefore also cost) to be spent on setting up, so it was vital to ensure there were no mistakes. The process of going from a written or typed work to the printed page involved various stages, including typesetting; all of which involved human labour, and where errors could be introduced. Before the final print was run, a 'galley proof' was produced. The proofreader checked this against the original document, looking for mistakes that had been introduced. In effect, this was a checking procedure only, and often did not even involve checking spelling, other that any differences between the final and the original. Spelling and grammar might have been checked by an editor, but frequently this was only the responsibility of the original writer.
By comparison, copy editing is more applicable to the digital age. Although it is frequently still referred to as proofreading, copy editing involves a great deal more. A copy editor will certainly check for spelling and grammatical mistakes, but will also look in-depth at the work, searching for inconsistencies, logical errors and factual inaccuracies. The best copy editors will understand the subject matter sufficiently in order to additionally comment on the actual content of the text. Last, a competent copy editor will read the text and suggest changes if applicable in order to ensure it is appropriate for the target audience. There is, after all, a substantial difference between an advertisement and a PhD thesis.