As has been commented on before, the state of the rivalry is such that the fans care far more than the players. The players have no particular reason to dislike each other, let alone hate anybody. On the Bruins' side only the likes of Coyle and Gryz were actually Bruins fans growing up, and they are too young to recall or to really grasp the full animosity between the two teams. Bergy, Krejci and Marchand were there for 2011 and 2014, but that's not a huge amount to go on and both teams' rosters have changed so much that it basically becomes just a memory from another era.
For the rest you can know the history, can want to represent the jersey, but unless you've either lived the rivalry and the passion for yourself or you've grown up in one of those families where it's in the blood, then you can't really feel it or understand it. The nature of the modern schedule and league only compounds the relative distance between the two opponents and between the past and the present. The players know these games mean more, but it's more on an intellectual level than a heart one.
For the fans of course it's different, although even then it's cooling, as it must. Still, it matters, and to not a few it matters a lot. So there is this slightly weird juxtaposition between the feeling in the stands and that on the ice. But it is what it is, and once the puck drops it won't change the fact that we are all that bit more desperate for Boston to win this one, and the excitement goes to that next level. Here's to a good game, and to the 22/23 Bruins finally winning a game in Canada!