What’s The difference in being lactose intolerant or being allergic to milk ?

Try drinking kefir for a while. There are different types of organism that can live in the gut, and some are not considered "resident", but are more like transient organisms that just move through with the food that you eat. They can be beneficial, but they don't hang around and help afterward. If you are so inclined, you can keep kefir grains and make fresh kefir every day. If not, you can find it in the grocery store.
Milk is one of the most adulterated foods we consume. Maybe you can find a particular brand of milk that doesn't make you sick. If you want to narrow it down a little, get kefir grains and make kefir at home. If your home made kefir goes over OK, then you are most likely lactose intolerant. The organisms that live in kefier, and yogurt, consume the lactose so that you don't have a response to it.
🙄Sorry, kind of wordy.