Editing and Proofreading are frequently used as interchangeable synonyms by those who don’t know the difference. Nevertheless, even if they are both part of the process of revision a piece of writing, they involve a different set of activities and happen at different moments.
Let’s stipulate that editing is for writing what hand planing is for carpentry, similarly proofreading in writing is like hand sanding in wood crafting. In fact, the hand planer takes the bumps out of a piece of wood, the sanding paper finishes surface and makes it smooth.

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Each tool has a function, you can use both or only one. If you edit your text, but you don’t proofread it, it will be like a flat wooden table with some splinters on it, oppositely if you proofread it without editing first, you will end up with a smooth and bumpy surface. If you want your text to be flawless you will have to do both and in the proper order: the proofreading has to go after the editing.

Editing

Sticking with our simile, the goal of the editing process is to remove bumps, i.e. improve the overall quality of a text, its readability and the flow of the writing. In order to do so the editor might change wording, make syntax smoother, clarify any ambiguity and cut down on repetitions. He might even reorganize the text, change titles or rewrite paragraphs. Of course, he also checks grammar and punctuations.
Finally, he can even prepare a style sheet, to make sure a manuscript follows a consistent style front-to-back (e.g. how to cite sources, how to use notes, how to use hyphens, formatting and so on).

Proofreading

Proofreading is like the finishing of the wooden piece. The proofreader waits until the work of writer and the editor is over, then his job is to read through the proof (hence the name proofreading) in order to sand out any remaining imperfection, i.e. typographical errors, minor text and formatting errors. Once the proofreading is done, the text is ready for publication.