In this specific case, there's little difference between the two - not only do they both mean "not now", they both mean "in the future". There's a subtle distinction:
"Let's talk about this another time!"
Here, "time" is used to mean an occurrence - so let's talk about this at one of the times when we meet, but not at this particular meeting.
"Let's talk about this at another time!"
Here, "time" means a time of day - let's talk about this at a different time today, or on another day altogether.
The distinction above may not be made by everybody, and either sentence could be read both ways unless you were being extremely picky about it.
There are some cases where the "at" would sound odd. For example:
- "One time when I went to the shop, there were no apples on sale. Another time there were no bananas."
- Not "
At another time there were no bananas."
In this example, "another time" means "a time (occasion) that's not the first one I mentioned".
A case where the "at" is required is where you're comparing with something that already has "at", meaning a specific time:
- "Let's meet at 3 o'clock" - "I can't - can we meet at another time?"
- Not "
Can we meet another time?"