I think a good way to frame the question is by contrasting product managers as strategic while business analysts are tactical. To clarify, I reword the statement: Product managers focus on multiple customers in a market. A business analyst focuses on multiple operations within a single customer.My original point was not to suggest that business analysis is necessarily tactical. On the contrary - business analysis can be tactical or strategic; it depends on the activities performed. The full business analyst role includes strategic functions, such as identifying business problems to solve and possibly even suggesting solutions. Companies, however, have a tendency to hire business analysts to perform only a narrow part of the full suite of responsibilities: documenting business processes. In such situations, "business analyst" is almost a misnomer; "documentation specialist" might better describe this very tactical role.
Though the responsibilities differ, product managers and BAs can both have strategic roles. The terminology that best captures the difference between the product manager and BA, in my opinion, is external versus internal. A company hires a product manager to analyze processes and problems outside the company. A company hires a BA to analyze processes and problems inside the company.