As you may or may not know, the musical is based on writer Jason Robert Brown's real life marriage...which failed. It's been like that ever since.
So, the likelihood of Jamie and Cathy getting back together after the movie ends is low, even though they are fictional characters. As viewers, we can see that their relationship was always meant to go to shreds, that they were pretty much destined to break up because we already knew the end result from the beginning. The first song in the movie is, after all, "Still Hurting," in which Cathy's simply being miserable after Jamie leaves her, and Anna Kendrick gets to show off her singing voice in a heartbreakingly beautiful manner. The fact that we know how the story will end makes the entire thing bittersweet.
From a directorial perspective, the last song says it all. Cathy, in a beautiful sunset, represents the beginning of their relationship when they were young and happy and starstruck. Jamie, on the other hand, is portrayed in blacks and blues, usually associated with sadness and bleak times ahead. He represents the end when they fall apart.
And, the last thing we see in the entire movie is a door closing. It's as simple as that.
There's no dramatic ending or held-out high note. It's just that door. It shows a finality that some viewers may not be willing to accept. It shows that the story for those two, together, is done.
Overall, it is a beautiful love story (despite the sad ending) because it signifies that not all relationships work out in real life. So why should they do that in Hollywood?