Proofreading is strictly-speaking the reading of proofs produced by a printer for a publisher.
There are usually 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 just the house-style: fact checking, spelling, and...
...convoluted, ludicrous statements that are overly and overtly (i.e. unintentionally or intentionally) verbose but entirely correct, logically, functionally and grammatically (or that hide their errors behind an unreadable barrage of words that readers are likely to skip-read).
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, a volunteer proofreader (friend or colleague of the author for example) can be a great help.
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. The return on my investment has been huge.