The main difference is what the systems are weighing. General search ranks pages, largely on relevance and on signals from other sites linking to yours. Local search ranks businesses, using three factors: how relevant you are to what was asked, how close you are to the person asking, and how prominent you appear, which comes from reviews, consistent information across the web, and mentions elsewhere. Distance is the one you cannot influence, which means the influence sits in the other two.
The practical consequence is that your business profile matters more than your website for local results. A complete profile with the correct categories, real services, hours, photographs, and a healthy number of reviews will outperform a better written website with a neglected profile. That is the reverse of the priority most businesses assume, and it is why local work often starts somewhere other than the site.
Consistency of your name, address, and phone number across directories carries weight here in a way it does not in general search, because agreement across independent sources is how these systems build confidence about a physical business. Small disagreements weaken it, and they accumulate without anybody noticing.
Where the two converge is content. A page answering a question specific to your area serves both, because it ranks as a page and reinforces relevance for the local query. That is the work worth doing when the profile is already complete.
Measure it differently as well. General search performance shows up as rankings and traffic, while local performance shows up as calls, direction requests, and profile views, most of which never touch your website at all. Your business profile provides those numbers directly, and reading them alongside your website analytics gives a truer picture than either does alone, particularly for a business where most customers act on the listing rather than clicking through.
Add pages specific to the areas you serve if you cover more than one, because a single page listing every neighbourhood performs considerably worse than a page about each with genuine content on it. The distinction matters: a page that merely names a place is recognised as thin, while one describing work you have actually done there is not.