I am with you on the Gary Jule's version of Mad World, Lust for Life. I think it's because the Tears for Fears original has the almost upbeat techno going on in the background so the line is almost skirted over and therefore barable. Whereas the slow piano version from Donny Darko leaves no escape for what is a brutally, honest, usually unspoken sentiment.
Context makes up a lot of whether something makes you cry or not. I was totally indifferent to the song "One day I'll fly away" by Randy Crawford. Then on the day I left my Mum's house, the family home realising I was never going to go back again, I got in the car and that was on the radio. Can never listen to it since.
I also get inexplicably teary when I hear La Marseillaise - usually when sung by women or children. That time they found explosives at that French football stadium and the crowd left defiantly singing it made me think just how rubbish our national anthem is and how little comfort we would get from it in a similar situation. All about one person instead of a nation.
The same goes for Nessun Dorma - I think I would be less moved by it if I actually had a clue what the words, other than the title, actually meant.