Product Manager usually is the business side of building a product. Which means a product manager needs to think about what market to serve, which problems to solve and how to solve that profitably. He does that by building a product which addresses the market problems with the right feature sets and marketing it to get a profitable outcome. In that process he is responsible for collaborating and directing a set of cross functional teams which could include marketing, sales, engineering, finance and project management.

Generally speaking Project Managers manage projects which have definite start date and an end date. They are typically responsible for managing resources, allocations, methodologies and schedules. Their ability to stay on scope, schedule, plan and budget enables a successful delivery of the project. Their teams would typically include engineering and business analysis resources.

Another wide observation is that you will see Product Managers on the product (software, hardware, engineering etc) side of the world, while Project Managers are quite often seen in services oriented businesses (IT Services, Consulting, Construction etc)

A simplification of these roles would be –

Product managers own “What” and “Why”.
Project managers own “How” and “When”.