Is there a difference in meaning between the two? For example:
- Carry on the mission
- Carry on with the mission
It really depends on the context, meaning, and message you are trying to convey.
"Carry on" IS the action.
I cannot believe the child would carry on like that in public.
"Carry on" on its own is a general command and is the same as "carry on with..."--one has less information and one has more.