Bring out the best in someone vs. get the best out of someone

No difference in meaning. Use whichever sounds more graceful to you.

I read to "bring out the best in someone" as meaning to get them to express their best traits, to be the best person they can be. "John and Mary's loving support of their children brings out the best in them. They are gracious and kind to all they meet." I read to "get the best out of someone" as meaning to get them to do their best work. "The coach really knows how to get the best out of his team. They've won the championship the last three years."

You're right. That's a worthwhile and meaningful differentiation.

I see bringing out the best as something that needs to be coaxed out, but getting the best not necessarily so. Any competent coach may be able to get the best out of professional sportsperson but only a particular approach or person will bring out the best in someone who's not there yet.