I feel like "Have a good one" is a bit more colloquial than "Have a good day."

In the U.S., these can be used interchangeably. "Have a good one!" is less formal. I would use it with a friend. However, I would say "Have a good day!" with an elder or a stranger.

"Have a good day" is usually said to someone who is leaving, like when going out of their home or a shop, or when ending a phone call. It's not said after evening/sunset. "Have a good one" is usually said to someone who is about to engage in an specific activity, like going to a sports match or a party.