The short answer
A social media manager is the person responsible for a business or creator's social media presence: planning content, writing and scheduling posts, tracking performance, and often responding to comments and messages. The role covers strategy and analysis as much as the visible task of posting.
What separates the role from "someone who posts"
Posting content is the most visible part of the job but not the most valuable one. Deciding what to post and why, based on real goals and real performance data, is where the strategic value sits. A social media manager who only executes a content calendar someone else built is doing a narrower version of the role than one shaping the strategy behind it.
The scope varies enormously by the size of the business
At a small business, one person often owns the entire role end to end: strategy, content, scheduling, analytics, community management. At a larger organization or agency, the role splits into more specialized positions (a strategist, a content creator, a community manager) working together, with "social media manager" sometimes referring to one piece of that larger team.