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.