Philadelphia - from our first playoff series through the redemptive conference final in 04 I think the Flyers were our biggest out of state rival, even if they never saw it that way. Regular season games against them always felt that little bit more important, more chippy, easier to escalate into chaos...more rough. Lots of oddly intermingled trade and player movement history between us and them, from gratton's offer-sheet-turned-trade all the way up to Vinny's CBO-signing. I love when we beat the Flyers. I can't wait until we see them in the playoffs again.
Washington - I think Washington took the mantle of our biggest rival in state or out over the last 10 years. Even this past season, not in their division, I cared much more about beating Wasington than Detroit or Columbus. I do expect this rivalry to wane some if the new playoff format sticks, forcing division opponents to play in the early rounds.
Montreal and Boston might soon become bigger rivals, but, other than a few years of overly inane Lecavalier fanstasies and the bitter 2011 conference final there isn't enough bad blood or competitive animosity between us and them...yet.