There aren't many "Program Managers" at Amazon. There are a lot of "Technical Program Managers". The T requires a CS degree and they are usually former SDE or QAE. They aren't usually compared to Product Manager, which generally requires an MBA, but there is a PM-Technical role that exists to scope features for platform teams and is common in AWS and similar but rarer in other parts of Amazon. Maybe you can clarify with example job titles from actual job listings and we can help you with better answers.
I am a Program Manager at Amazon. I think most of the responses here are in the context of a tech feature - product managers plan roadmap etc and program managers manage execution. On the business side, it is different and the lines are quite blurred. Both product and program managers manage one aspect of the business, say pricing. The distinction is involvement in the day to day business operations. A program manager is more involved in running the day to day business than product managers.
How do the two compare in terms of pay and impact/influence in a project? And growth in the company?
Program Manager = Project Manager Product Manager = well... Product Manager. Project managers are doing schedules, wrangling meetings, glorified admins really. Product is figuring out what's best to build. Long term strategy. That kind of stuff.
I assume you came from Amazon? Since this question is specific to Amazon.
It's like this basically everywhere in the industry besides Microsoft. Microsoft has Product Managers but what they really mean is Product Marketing manager. They have Program Managers, which sometimes means 'Process/Execution PM' aka Project Manager and sometimes 'Feature PM' which is more like a Product Manager everyone else. Shitty for you, this makes it really hard for Microsoft PMs that enjoy doing features to switch companies and not wind up project managers.