When someone tells you to ‘get on with it’, they are suggesting (or ordering) that you start or continue a specific task. If you are lazing around and your boss is angry at you, he or she might say ‘Get on with it’, indicating that you should stop doing nothing and start working or get back to work. You can be specific, and say ‘Get on with your homework’ or ‘Get on with that task I set you’.
When you say you ‘get on with someone’, it means you connect socially and find if easy to talk to them. If you are relaxed around a person and you don’t get ‘lost for words’ in their company, you ‘get on (well) with’ that person. You can also say that you both ‘get on well together’.