It depends on the context. Say you are on a road trip and you stop for lunch. Afterwards, you say to your companions: "Time to move on" (to the next place on the road) or "Time to carry on" (with the road trip). Those would mean the same, effectively. However, suppose you ask your boss whether you should keep doing some task that you suspect is running out of steam. They might reply either: "Yes, carry on", or "No, let's move on to something else". Symon's example would have that meaning, too. There's also an idiomatic meaning of "carry on", which I think is mostly British, meaning to misbehave.

i'm English learning too, but base of my understanding "move on" is just like saying to stop your self, to stop being obsess of, or something. example "you love someone so much and then you saw him with the others, and you felt so much hurt for this so then you've realize to move on." move on you feeling from someone or to stop your feelings... i hope it could help... :)