What's the difference between a product manager and a product owner? These terms are often used interchangeably in job postings, so it can get a little confusing.
A product owner is a role on an Agile team; product manager is the job.
However, this isn't necessarily how employers use these terms. These terms are a bit different, and here is my take on what they mean, based on my experience.
Product owner: "Product owner" is an Agile-specific term for the product role within a Scrum team. In an Agile environment, the product owner is responsible for defining user stories and prioritizing and maintaining the backlog (this is really stressed in Scrum). The PO represents the interest of the customer on the team. In my experience, PO's spend their days in all Scrum ceremonies alongside the dev team(s) and are really into the details of user stories, clarifying requirements for devs and QA, and very integrated into releases/ demo-ing at the end of sprints. i.e., if a dev needs to know "should this button say "Buy" or "Purchase"? - they'd ask the product owner and the PO will hunt that answer down. In general, I think of a "product owner" as a Scrum-specific role.
I've also heard the term "product owner" refer to my superiors as the true "owner" (read: decision-maker) for the product. IE: The product manager works "in the trenches" every day, writing user stories, collaborating with the dev team, etc. However, a Senior Manager or VP may be referred to as the "product owner" who calls the shots, when the shots need to be called. In my opinion, this is an incorrect use of the term, albeit a common misuse at many companies.
Product manager: A "product manager" role is more broad than a product owner. Although a product manager's priority is also gathering requirements, requirements may be in forms other than, or in addition to, user stories. And, while a product manager may be involved in prioritizing the backlog (often in the form of defining a roadmap), he or she may also be involved in higher-level product strategy, product marketing, pricing analysis, and end-to-end planning. In general, I think of a "product manager" as having a wider breadth of responsibilities than a product owner, and the team doesn't need to be running Scrum.
Other roles: I've held the title of "business analyst" before when I was really playing the role of product owner. Lots of companies nowadays are hiring BA's to focus on writing detailed requirements (often in the form of user stories, epics, etc.) specifically for developers, while product managers focus on the higher-level strategy. Oftentimes, BA's work alongside, or under, a product manager, acting as their requirements-writers, so product managers can focus on the bigger picture.