Looking at n-grams, it seems that before 1910 "help on" was standard, and since then "help with" has rapidly become a lot more common. There seems to be almost no difference in usage frequency between British English and American English and I personally can't think of any instances where "help with" or "help on" can be used and the other can't. Sometimes "help on [an object]" might be a bit ambiguous because "on" might be taken to suggest that the help takes place on top of the object.
Do you want some help with the roof?
makes it clear that the purpose of the help has something to do with the roof, but can't be read to specify where the help takes place, whereas
Do you want some help on the roof?
could mean the same as the previous sentence, or it could mean that I'm offering to give help which will happen on the roof, but I'm not saying what the help is with (maybe I'd be helping to rescue a cat - so I'm helping 'on the roof', but not helping 'with the roof').
So (apart from possible ambiguity with 'on') I think they are more or less equivalent, but "help with" sounds a lot more natural in the present day. (Though I wouldn't be surprised if there is some regional variation with this).