Far and away, the best rivalry in New York sports (before any Buffalo fans message me, I'm talking city) is far and away the Islanders and the Rangers.
The fact that the Islanders and Rangers are in the same division/conference makes all the difference in the world. Unlike the other rivalries in the city, at least till the Nets move to Brooklyn, the Yankees/Mets and Giants/Jets just don't play nearly enough. As it is the Giants and Jets only play a regular season game once every four years. The Yanks/Mets play two series a year, but in the big picture of the insanely long MLB season six games doesn't mean much.
When it comes to the Islanders and the Rangers, it's six battles a year and the much more likely possibility of meeting in the playoffs. Not only that, but from Day One of this rivalry a $4 million territorial fee to the Rangers just screamed that it was going to be a mutual hatred. The first playoff series the upstart Islanders ever won was in the old preliminary best-of-three series against the Rangers. During the Dynasty years the Islanders met the Rangers four times in the playoffs, winning every series.
The history is just so much richer due to this.
The "Potvin Sucks!" chant and 1940 chant both have their respective places in history. It's been 20 years since Denis Potvin played in the NHL. Imagine if a Mets fan tried to start a "Mattingly sucks" chant at Shea or Citi field?. He'd get a bunch of odd disinterested looks. Or if a Jets fan tried to start a "Simms sucks" chant, you can imagine it would be more of the same. But with yearly games, and wins against NHL divisional opponents at a premium, the year-to-year wounds never heal. That's why any of the other pro teams in New York would be hard pressed to reach the enmity at an Islanders/Rangers game.
Given the small amount of playoff victories in my history of following the Islanders, it seems like the games I always remember the most were wins over the Rangers. Growing up around 90% Rangers fans is probably part of the reason for that. That the Islanders were terrible and in a way the Rangers were responsible (take this Maloney guy, he's great!) probably had something to do with it. The Islanders always seem to find a way to play the Rangers tough, even when they are incredibly awful.
Compared to Islanders/Rangers games, the Jets/Giants and Yankees/Mets just don't carry that much weight. If your team wins and you have a friend or relative whose a superfan of the other team it's a bit of fun (my stepdad has Yankees everything, he's probably part of why they can afford that damn roster) but in the end inconsequential. When the Nets move, sure the Knicks and Nets will be in the same division, but it will take a long time for the history to build up. It'll also take a long time for either team to be good.
In the end it seems like we all have our own little history or story about the Rangers as Islanders fans. From the loud mouths who seemed to every year predict that this free agent would be the one to get the Rangers the cup, to the ones who claim that Snow will be riding the Zamboni too. I'll always remember a game from the 95/96 season. The Rangers dominated every game that year except one. In the process they put up 4 or more goals in every matchup. The Islanders were just two weeks off of a home and home series which the Rangers handily won 4-2 and 6-2. At the time the Rangers were also on a long unbeaten streak at home.
To make matters worse, the "best" Islanders were all out hurt, including Ziggy Palffy's one missed game on the season. To make matters even worse the goalie was Tommy Soderstrom. But the Islanders fought on behind Marty McInnis (2 Goals, 1 Assist) Wendel Clark (2 Assists) and Martin Straka (1 goal, 1 Assist) to upend former Islander Glen Healy and win 5 to 3 at the Garden. I got to hold my head up high at school for at least a day or two.
The closest any of the other NY-based rivalries can get to that would be the Subway World Series. It probably has a bitter aftertaste due to hindsight and the Mets never getting back to the World Series. We all though have the same stories on Ranger fans or Ranger games we remember, feel free to share them below. With the second game of the season Monday being a showdown with the Short Island Smurfs, we might as well get ready early.