Q: Change Apple ID

You can't switch it.  You can use them interchangeably as they both serve as your Apple ID.

See this -- iCloud: About your @icloud.com, @me.com, and @mac.com email addresses - Apple Support

About your Apple ID email addresses - Apple Support

You won't be able to delete the @hotmail email address as that serves as your Apple ID.  You can change it, though, to another non-Apple (@me, @icloud, @mac) email address.

I don't understand how/why your "apps, backups, and even iTunes credit" would be lost.

Yes, your account balance, etc. will be untouched.

You could change the @hotmail address to @yahoo address if you wish as your Apple ID.  See this for further reference -- Change your Apple ID - Apple Support