No, they don't have the same meaning.
1. 'Another time' means 'not at this time'. So if you are too busy to speak at the moment, you might say:
'I'm busy now. I'll talk to you another time.'
or, more politely, you might say something like this:
'I'm a bit busy at the moment. Can we talk another time?'
2. 'Again' refers to a second or repeated occasion ( 'un'altra volta' in Italian). So, at the end of a conversation, you might say this:
'This has been a very useful meeting. We must talk again some time.'
I hope that makes sense.