Cut away and cut off In many cases (as this one) they can be used interchangeably: you could also "cut off" the parts the dog didn't touch. "Cut away" has a more specific meaning of removing something that you don't want, whereas "cut off" is more general (you are simply removing something by cutting it, whether you want to keep it or not). "Cut off" also has a set of figurative meanings (e.g. you can cut someone off, meaning interrupt them; you can cut off supplies, meaning stop them; you can cut someone off, meaning stop being their friend).