It's actually also used in informal settings among people of the same rank and not necessarily only as described as above. It's also used as a friendly and gentle invitation and sometimes just as a simple verb in the indicative. It's not always used as an order with people of different rank and yes, it can be interchanged with 'continue' in some circumstances. It's quite diverse and there are actually more examples with 'carry on' and 'carry on with': "Carry on with what you are saying. I love listening to what you have to say " = continue with what you are saying . . . "If you carry on in such an immature/childish/improper way, I am leaving the room" = If you continue (acting/behaving/talking in a certain way), I will leave . . . "Carry on with what you're doing. I'll be back in a minute" = Continue with what you're doing . . . "If you carry on with him/her, we will have to part ways" = If you continue seeing/being/interacting/being in a relationship with someone, we can no longer be together. "If you carry on eating junk food, you will get more unhealthy." = If you continue eating . . . So 'carry on' can mean 'continue' and it's 'carry on with + person' or 'carry on with + verb"

Carry on is used as an order. Usually by a superior to a junior to continue whatever they were doing when they stopped to acknowledge them. This is usually used more in the military setting. A friend will also use this casually in a joking manner with another friend. Jokingly because again it is usually an order from a superior. Carry on in another casual sense means to "/get over emotional about something in a dramatic way." An example is: You don't really need to carry on like that since he dumped you. Since carry on is more an order it is not usually easily interchangeable with continue. To carry on with - specifies continuing the over emotional drama with another individual/or some topic. example: You really don't need to carry on with this conversation. I said it was over already. example 2: I don't even see why you carry on with him like that. Just let him go.