This isn't quite true, but depends a lot on play style IMO. We play with time-constraints for most of our adventures. You can't start it and then take two months off without issues arising.
Take our current adventure. We are infiltrating a stronghold, moving through it, and taking out the bad guys as quietly as possible. We know we aren't going to be able to finish this before we have to retreat. Depending on our status, that retreat will hopefully just be a long rest. But, if someone gets injured badly, we might have to leave them behind the next time we go in since they won't be full strength. Or, maybe they'll come along but be weaker and we have to be more cautious.
Rests, and the time in between them, can certainly impact the game play.