An instant is a very short time. In combinations like instant coffee and instant replay it means available right away without a wait.
Instant and moment mean the same thing if you are talking about a particular time in the past like you remember the instant or moment you realized that you had won the lottery. However, the two words are very different when talking about time in the future. "I'll be there in an instant" means you're coming as quickly as possible, whereas if you say "I'll be there in a moment," someone is going to have to wait a bit for you.