*I am not a teacher*
Hi - the way in which I would use them is to do with what I think is called 'verbal aspect', which concerns the way in which the speaker views a process. I have heard it described as when you look at a street carnival, if you are a reporter viewing it as a whole from a helicopter, you would get a very different view from a reporter on the ground. One is viewing the carnival as a whole, the other is viewing it from inside.
"He is obsessed by dates in history" - I would use this when I am describing the person, but I am viewing this as an observer from a distance. I do not feel any particular emotional connection with it; it is just a cold fact.
"He is obsessed with facts about dinosaurs" - when I use this, I am more 'inside' the experience of the one who is obsessed. As I say it, I may visualise them as sitting with their head in a book, reading avidly, and it feels more personal, as if I am actually experiencing the obsession and identifying with it more.
I also wonder if 'obsessed by' is more likely to be used in US English?
Ade
*Not a teacher*