Mostly it is a difference in style and tone. "Intestine" or "intestines" is a polite, fairly precise, medical term. It includes both the large and small intestines, but not the stomach. "Bowel" or "bowels" can mean, vaguely, the gut or anything in the abdomen. It can mean specifically the intestines. But it most commonly means the large intestine or colon. In theory it is perfectly polite and neutral. However, we use the euphemistic phrase "bowel movement" as a polite way to refer to the act of defecation or to feces themselves, so the word "bowels" is somewhat unpleasant. By the way "bowel" rhymes with "owl" and "howl" and has the same vowel sound as the o in "how" or "now." "Guts" refers to the whole digestive system--stomach, intestines. It is not a bad word, but it has a negative connotation. It carries the idea of these organs _after death._ For example, "traditionally, sausage casing is made from the guts of an animal," or "traditionally, violin strings were made from catgut." It also carries the very unpleasant idea of a terrible injury, as in wartime, where "blood and guts" spill out of a body. "Guts" are the informal word for anything inside your belly that you feel. You can say "I have a pain in the gut," which means the same thing as "I have a bellyache." Both are informal. (A doctor would call it "abdominal pain.") Intuitive ideas are called "gut feelings." "I know the forecast is for sun, but I just have a gut feeling that it is going to rain." Finally, "guts" can (curiously) mean the same thing as "nerve." Both can mean "audacity." "I'm going to walk right in there and demand a raise." "Wow! You've got guts! Good luck!"

These terms are used interchangeably, which means they are used to describe the same thing. However, they are slightly different. The gut is a slang term, typically used to mean the overall organs in the stomach area. The intestine is used for a particular organ of which is broken down into the smaller intestine, and lower intestine. The lower intestine is also known as the colon. The bowel can be used to describe the intestines if you call them bowels, though, I believe that it's mainly used to describe the lower intestine/colon. I hope that helps!