Depends.
As a f'rinstance regarding films, I'm currently watching Sicario (September 2015) , while Mrs M watches Brooklyn (Jan 2016). The rule of thumb, I think, is Netflix for TV, Prime for films, although that holds true only so long as the things you want to watch are present. There are enough differences between the two for us to justify both services, although we don't pay for Sky, BT Vision or anything like that.
Netflix has a lot of very good original programming (there's a new David Cross special they've just commissioned that I'm keen to watch, there's Stranger Things, Breaking Bad, Narcos, Marco Polo...) but Amazon have some decent offerings too (Vikings, for instance, is oddly compelling, Transparent was very good indeed). The other issue, of course, is that if the distributor has negotiated an exclusive with another service, you'll never see it on Netflix or Prime. (Game of Thrones, I'm told, is reserved for HBOs own streaming service - The Walking Dead does seem to come to Prime, but with a long wait indeed).
If you absolutely have to watch things as soon after release as possible, you'll need to either pay to stream, or buy the Blu-Ray.