Consider the definitions of each role.
Entrepreneur: a person who organizes and manages any enterprise, especially a business, usually with considerable initiative and risk.
Project Manager: [has] the responsibility of the planning, procurement and execution of a project, in any undertaking that has a defined scope, defined start and a defined finish.
An entrepreneur can be a project manager, but not vice versa because the project manager is typically only risking assets of the company. Entrepreneurs will risk personal assets e.g., money, property, relationships, etc.
It's difficult to define the key tasks of each role because they differ depending on the industry and organization. I have had the experience of being both, and in both roles I've been required to have knowledge and skills of every position in the organization.
I will say that the defining factor for me is this: time.
As an entrepreneur, I spent every waking hour consumed by my business and obsessed with exceeding the venture's goals. When I was project manager, I clocked in and clocked out with occasional after-hours work here and there. I took days off without any concern for how the company or business was functioning without me. I can't say this is the same for everyone, though.