The word "differing" suggests some contrast; they "actively" differ from each other. There is some sort of disagreement, some sort of connection.
"Different" merely means "not the same" but there isn't necessarily a contradiction.
"Our cultural backgrounds are different." - this means they are not the same but doesn't suggest anything beyond that. Maybe the two cultures have no problems with each other, maybe they do, it's unsure.
"Our cultural backgrounds are differing (from each other)." - this means they are not the same and there is some form of disagreement. They don't want to be like the other.
This is how I see it. I'm not a native speaker, hopefully someone else can explain it better. Hope this helps though.
("on a personal level" can mean privately.. it means that it is related to and afftects only one or a few individuals directly)