People who excel at both product and project management are often compared to unicorns because they are so rare. The responsibilities are different enough that people tend to neglect one set for the other and, unfortunately, it is usually the important but less-urgent research, strategy, and prioritization activities of product management that suffer.
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Given this, hire a project manager if you need someone to ensure effective delivery of work products, including the following:
Planning
Scheduling
Cost management
Risk management
Hire a product manager if you need someone to drive the success of a product through its entire life cycle, including the following:
Understanding market needs
Developing and communicating product strategy
Prioritizing investments
Developing requirements
Supporting go-to-market efforts
Measuring business results (and using them to inform future plans)
I dive deep into this topic in my book Product Manager vs. Project Manager. Agree? Disagree? Tell me a story..