Ha, I never actually paid attention to that. I think the fact that I'm not a native English speaker could be why "dampener" never sounded weird to me.
Although looks like the Oxford English Dictionary says both have the meaning "thing or person that has a restraining or subduing effect", so I guess they'd both be correct in that sense?
Through idiomatic usage, dampener has taken the same meaning as damper, but as my dynamics professor was fond of saying, one of them terms means to slow motion, the other means to get something wet.