I like songs that have sad lyrics and happy tunes. For one thing, the contradiction is beautiful. It indirectly gives the message of finding a way to be happy no matter what the circumstances. Also, I think this type of song has a way of putting a sad or broken hearted person in a good mood. The lyrics give you that feeling of connection, like the song is describing the way you feel. At the same time, the music itself moves you into a completely different state of mind. The song that made me think about all this is "Bye Bye Love", which I found out is actually by the Everly Brothers, but I'm used to the Simon and Garfunkel version.
A similar phenomena are pieces that express intensely sad or even angry emotions at the beginning, then, by some believable miracle, it becomes hopeful and encouraging. It's like the happy/sad songs in the sense of establishing a connection with despair and transforming into something beautiful.
The difference is that it's more likely to fully express the pain, letting it run it's course before moving on. I guess it's the music aspect, more than the lyrics, that I truly relate to...which is probably why sad songs with happy lyrics just don't do it for me as far as getting into a better mood goes.