As for reasons why I like those kind of movies, is probably because they relate to the existential crisis that hits me every now and then. My favourites among this genre are "About Time", "Never Let Me Go", and "The Fault in Our Stars". Even "Up" made me feel a little melancholy. Yes, I'm aware of how they are borderline depressing, but I find that they also provoke thoughts about the meaning of life and love (note: not romance).
On a side-note, I've always thought that I'd be a clinically depressed person if I wasn't Christian. The hope that God gives not only makes Him worth dying for, but also to live for.
And with that I present to you the tune that is so sad it'll make you cry your eyes out: