- I don't care
This indicates complete detachment and disinterest. It means that something is not important at all. You can do what you like, I don't care, it is nothing to me.
So, watch out! It may sound impolite depending on the context.
- I don't mind
This is a gentle statement indicating general assent. It is a quite polite way of expressing you have no preference, that something doesn't bother you, that you give your permission.
Now that you already know the difference between these two expressions, be careful while using them because they don’t mean the same.
Hope you like it! 
Teacher Nathalie
