I think the timing of each team's runs matters, even if the net effect is still a similar record.
With Washington, we more or less had a decade of regular season dominance while never getting past the second round, and then "all of a sudden" one year with the same core in place they put it all together and win the Cup, albeit in convincing fashion. They follow that up with a first round exit the next year. I think it's foolish to label it as a one time thing where they just put it all together just a year after it happened, but the point still remains that we've only ever seen one good run from them.
With Boston, while they've had that core of Bergeron, Chara, Krejci and Marchand in place since basically the start, they've iced 3 different teams that have gone on runs, all at very different points. A lot of the talk about the Cup win is that they rode a hot goaltender to the title, and then they come back a couple years later with a completely different goalie and make it to Game 6 of the Cup Finals. After a couple years of retooling, shaking things up and bringing in the next wave of players like Pastrnak, DeBrusk, Carlo, and McAvoy mixed in with the old guard, they're primed for another Cup run. I think this year regardless of whether they beat Carolina or not, winning multiple series in a playoffs classifies as a run rather than a fluke.
I think the main point is that with 2 teams that have held onto their core players WAY longer than most other franchises do, we don't really know if Washington's run is just a one-hit wonder, where as we know that the Bruins have proven they can do it multiple times.