The short answer
An SEO consultant grows the qualified traffic and revenue a website earns from search. They do that by making the site easy for Google to crawl and trust, building pages that match what your customers search, and earning the authority to outrank competitors. The good ones translate all of that into business outcomes, not jargon.
A consultant differs from an agency in one way that matters most: with a true consultant, the senior who diagnoses your site is the one doing the work, not a sales lead who hands you to juniors.