I do not agree they are totally equal, or that one is merely Br. and one is merely Am.
While both expressions can be used for people, in North America, you could use "get on" like this: How are you getting on with that book/project? "Get on" here refers to how much progress you are making. Is your progress rapid or slow? However, it would not be typical to say, "How are you getting along with those shelves." This is an oversimplification, but you get along with people, and you get on with an action/activity/job/hobby/project.