Proofreading is strictly-speaking the reading of proofs produced by a printer for a publisher. There might be up to three proofs and proofreadings, depending on budget, schedule, and quality standard.
Usually, at least one proof is also sent to an author for checking before the final printing.
The proofreader indicates changes that are needed, and then a copy-editor (close to the publisher) may make changes needed for layout and the publication style, before giving the text back to the printer for the printing of further proofs, or for final printing.
Copy-editing can be regarded as a specialised form of proof-reading. General proof-readers have wider concerns than the house-style: fact checking, spelling, and...
...unintentionally convoluted and ludicrous statements that are overly verbose but entirely correct, logically, functionally and grammatically. I hope I spelled that right.
In many situations, the proofreader is the author, and the copy editor is the managing editor of the journal.
When the publisher is small and lacks in-house staff, then it is great if a volunteer proofreader is available (a friend or colleague of the author for example).
If funds are available to pay for a professional proofreader, this can be very useful, for both the author and the publisher.
I paid my house-mate in Australia something like $400 (AUD) to proofread my 400 page PhD thesis in 1990, Canberra, Australia. Rates vary.