For Americans, there is a subtle difference. If I got through something, it would mean that I experienced it and am finished with it as far as dealing with it directly. "I got through finals last week OK." "I got through the vacation with only a sunburn." If I get over something, it means that it no longer affects me, either physically or mentally. "I got over the flu last week." The best way to explain the difference would be if you had a bad car accident. You might get through the accident with minor injuries. It would take you longer to get over (completely heal from) the injuries. Also, you could have nightmares about the accident, in which case we would say that you got through the accident, but you haven't got over it.

I think it would be "get over", but that's just personal preference talking. Either one should be fine, in my opinion. There's no hard and fast rule about this but you get THROUGH a break-up, and then get OVER the ex-boyfriend, get THROUGH a severe illness, but get OVER a cold. Basically, get over may be used for more trivial things? Don't quote me on that, though :D