A lot of times to get on refers to getting on a bus or another type of transportation. However, you will hear to get along with someone more often in terms of relationships. For example, to get along with your friend or to get along with your family... I hope this helps.

Yes, they do mean the same thing. 'Get on with' with is heard more in British English and 'get along with' more in American English, but both expressions are used and understood by all English speakers.

Thank you Salsero, I'll bear it in my mind

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.

It's sort of the difference you find in 'sit on' and 'sit upon'.