Change your mind means that you change your opinion about something:
When I first started reading 1984 I wasn’t sure I was going to like it. I thought it was a bit slow and too depressing. But I soon changed my mind once I got into it and now I think it’s a fantastic book.
Change your tune similarly means to change your opinion, or your behaviour changes. We use it slightly differently to change your mind. We usually use change your tune when the opinion or behaviour goes from a negative to a positive.
Imagine you have a friend who has been moaning about her boyfriend for weeks. Really negative about him … they are arguing and they are not in a good place. Then suddenly, your friend starts saying how amazing her boyfriend is, she’s really in love with him, he can’t do anything wrong ….
In this context you wouldn’t say to your friend: You’ve changed your mind!
You would say: Well, you’ve changed your tune!