I've never truly considered the Jets a true rival in the same regard as I did the Dolphins in the old days, but I've always hated the Jets more than almost any other team. Part of it is the NY aspect where the players, fan base and media all hype them up far beyond is appropriate due to the market. They aren't now nor have they ever been the NY Yankees, but you wouldn't know it listening to their fans' sense of entitlement. We saw the media hype ramp up when Rodgers was a Jet, though it was proven to be completely unwarranted.
They were competitive and entertaining during the Rex Ryan years, but during my lifetime, they've almost exclusively been pretenders. The Jets haven't been to the Super Bowl since Joe Willie Namath, for crying out loud.