Service Workers (which are one of the core technologies behind the Progressive Web Apps label) go further than manifest files because they can deal with any kind of HTTP request (including e.g. calls to your REST API). Also, because they are code rather than static lists backed by hard-wired browser logic, you can implemnt any kind of caching strategy you want.

That being said, Progressive Web Apps is pretty much just an umbrella term useful for marketing several new and upcoming browser APIs (Service Workers being one of them) and not really a thing in itself.