They mean roughly the same thing, nagomi. "To obsess over something" sounds a little more active than "to be obsessed with something".
I would say "obsess over" is informal, mainly because that's how I've heard people use it. I don't recall hearing it years ago so my impression is it came along after "be obsessed with." In my 1970 AHD it says "obsess: 2. to harass or beset like an evil spirit; haunt as a fixed idea." I believe "fixed idea" is an older psychological diagnosis. The "obsess" usages are probably more recent.
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"To obsess over something" has more of an American flavour to me. Australians tend to use "obsessed with/by something".
Americans definitely say it. It may be said in BE as well.Does over really sound American to you?