Then I guess the best NFL rivalry would be the Bills and.............uh, the Bills and.............Bills and....... NOBODY cares about the Buffalo Bills.
I think Pats-Jets is a big rivalry still. As long as Belichick is there - maybe longer. The history with Parcells, BB, the players... etc? Case in point, I'm a Pats fan that likes to see what the Jets are up to - I also consider those two games to exist independently of what's going on with the rest of the season. 2-14 could be a lot better if those 2 wins come against the Jets.
The bills don't have a current rivalry. Neither do the jets, or the dolphins, or the patriots. The closest I can think of that borders the term rivalry is maybe pats/ravens. That's about it.
Maybe Tannehill and Luck are developing a nice little rivalry after 2 thrilling games the last 2 years ... but for the last 10 years IN THE AFC EAST, I'd have to say that everybody is wanting to be the one to put down the pats and be the true thorn in their side. The Jets have had the most success in that regard IMO including the playoffs. Just as in the early 90s, everyone took their best shot at Jim Kelly's Bills.