These are all pretty much interchangeable, but there are indeed some nuances: You usually "give up" things that might be seen as indulgences, like smoking or too many desserts. - Do you really live longer if you give up smoking and eating desserts, or does it just seem like it? You cut out things from a larger group of similar things, like gluten from your diet. With habits like smoking, "cut out" is a little dated now. You still cut out behaviors. - I feel a thousand times better since I cut out eating gluten and high fructose corn syrup. - Man, I cut out smoking around the time I stopped going to Vietnam War protest rallies. - (in response to the children doing something they shouldn't) Hey! You kids cut that out! You might also "cut down" on things that you're going have less of, like cholesterol from your diet. - No more bacon cheeseburgers for me: I've got to cut down on my cholesterol. (I think it's a mistyping, but while you can "cut up" an unhealthy food, that just means using a knife to make it into smaller pieces of unhealthy food.) You can give up, quit, or stop an unhealthy behavior. These are just about interchangeable, but in order of very slightly decreasing formality: - Mr S, as your doctor, I must advise you to give up smoking. - The doctor told me I need to stop smoking. - Am I ever gonna be able to quit smoking?

I would say that they are all synonymous, and you could use them all with food and smoking. I would also say that 'quit' sounds very American, 'give up' very English, and 'stop' is very dry, standard English.