Proofreading is the process of checking typeset proofs against the original manuscript in order to correct typographical errors, mainly in spelling and punctuation. The proofreader checks that all headings are correctly set and that the publisher's style sheet has been followed.
The proofreader indicates errors using standard proof correction marks. The proofs are then returned to the typesetter for revision and a further set of proofs produced. This stage is the final check before printing.
Copy editing is a much more fundamental process where any problems are solved before the manuscript is typeset (thus minimising expensive proof corrections). Copy editing entails correcting spelling, punctuation and grammar; checking consistency and continuity; flagging up quotations that may need copyright permission; verifying cross-references; and raising appropriate queries with the author. It ensures that the text is complete, with all components clearly identified and with a style sheet prepared for the typesetter and the proofreader.
The copy editor ensures that the edited document conforms to the publisher's house style (or a self-publishing author's own design), with standardised headings, sub-headings, footnotes, bibliography, etc and perhaps creation of an index, where appropriate. It may also be necessary to abridge or rewrite text either to fit a given format or to improve its flow.