so many of the so-called 'genuine' batteries sold on Amazon (certainly on ebay) and other places are fakes, that I think the main difference is it's almost worse to buy a supposed 'genuine' battery unless you're 100% sure it's legit, vs. a decent 3rd party make like Anker.
On amazon, read user reviews. So much of the OEM stuff are total fakes, almost guaranteed if they cost significantly less than usual. On ebay where there's no direct product feedback, assume ALL are fakes.
I've found no significant difference with a decent brand like Anker, and the original that came with the phone. (Note 2, 3 and 4).
The main things to look for are decent user reviews, a realistic price (if it's $6 instead of $16 for example, it's fake) and
if you need NFC, make sure it's supported.