There're actually more flavors of management role than just those two, and the terms get a bit confusing when you try to map them:
1.) Tech lead. This is the technical point of contact for the project as a whole. Responsible for deciding overall system architecture (usually by acting as the tiebreaker when there're debates within the team), determining priorities, either assigning work or creating systems where the team can identify what needs to be done and do it, interfacing with their PM, pitching in technically when needed, and ensuring the overall engineering success of the project.
2.) Engineering management, or more generally, people management. This is what's reflected in the official org chart. Responsible for career development of subordinates, keeping reports happy and ideally still employed at Google, dealing with personnel issues, HR issues when necessary, managing performance, coaching, giving feedback, getting their reports promoted, stepping in when the employee has done something inappropriate, etc.
3.) Product manager. Drives the product vision of the project forward. In theory (and hopefully often in practice) they represent the user and are responsible for building something users want. Daily duties include negotiating with other functional areas of the org (legal, PR, security, privacy, accessibility, etc.), doing market research, analyzing data, representing the team in meetings, keeping the product vision, coming up with new ideas, often building mockups (although this is often outsourced to UX), ordering T-shirts, getting approvals, and generally doing everything necessary to get the project to launch.
4.) Program managers. These are process-oriented managers that are often attached to infrastructure or devrel teams that have ongoing responsibilities that impact a number of consumers. Their job is basically organization and communication: they make sure everyone knows what they need to know and that all external requests are eventually handled with nothing important being dropped on the floor.
#1 and #2 are both done by engineers, although #2 is sometimes done by dedicated engineering managers. #3 and #4 are specific job titles. Sometimes #1 and #2 are the same person, sometimes they are separate people; many people find things run more fairly and more efficiently if technical decisions are not made by people with direct authority over your career, although others prefer to have their manager be intimately familiar with their work because it helps come promotion time.
#1 and #3 also map to startups, where a "tech lead" is basically the "hacker" while a "product manager" is basically the "hustler". #2 maps to the "adult supervision CEO", while #4 maps to "super talented admin who keeps the books, knows all the employees, and keeps the company running."
