It’s simple: Branding is what you use to mark your cattle. You take a red hot iron with your logo and burn it into the hide of each cow or steer you own.
PR is what you use to explain why you are burning the poor, innocent cows. PR explains why this is important, PR explains why it really does not hurt the cow, etc. PR, if it is good PR, convinces people that it is fun and the cows love it.