In case it's not clear enough, the phrase is usually said by someone who has had a new, happy experience on the day. Suppose you fell ill and had a lousy day, but it almost went away completely the next day. You'd be pleasantly surprised and might say "What a difference a day makes!", to mean you feel so much better, i.e. things are so different, than the day before. People don't say this phrase like a general proverb to mean that everyday is important or a lot of things happen in a day.

No, it's more idiomatic. It is an expression that expresses that a lot of things can happen or change in a day. Or that things can rapidly change during such a short period of time.