I guess I'm saying the latter too. Can you explain the former?

It is my understanding from reading Maynard's preprint that he claims independence from Terrence Tao and the Polymath8 project. He doesn't claim independence from Zhang and neither should he, even if he uses a different approach. His result may well turn out to be important or more important but Zhang's result is clearly earlier and well-known.

My differentiation is that number theorists are looking for better sieves all the time. Zhang's work was a breakthrough in bringing the prime gap from infinity to a finite number. It is not surprising that better sieves could bring the gap a lot closer.