The use of "gross" in colloquial speech... as in "Ewwww! That's gross!" started in, I think, the 1970s or 1980s. "Gross" is colloquial and informal. In the word "disgusting," the "gust" part is the same root as "gusto," "gustatory," relating to the sense of taste. In its most literal sense, it means something that is so unpleasant that it spoils your appetite and makes it difficult to eat. The word "gross," in its literal sense, simply means "big" or "a large number." An old meaning of the word "gross" is a dozen dozens, 144. The origin of the word "grocer" is "a storekeeper who buys things by the gross." However, it carries the idea of too big, too fat, corpulent--and the idea of "coarse." It's just a wild guess, but I've wondered whether the modern meaning is influenced by the term "gross anatomy." "Gross anatomy" literally means the anatomy of the major organs of the body, the large-scale anatomy, the subject that medical students study that involves dissecting entire human corpses. Non-medical people find that disgusting, and it is easy to see how someone hearing the phrase "gross anatomy" could guess at the meaning and decide that "gross" means "disgusting."

They're the same, except 'gross' is more slang/informal.

In some cases, they have the same meaning and can be used in the same way - particularly if your disgust is 'physical'. If you are describing something that looks, sounds or smells so horrible that it makes you feel ill, then - informally - you can say either 'gross' or 'disgusting'. However, 'disgusting' can be used in other contexts. You can describe a situation as disgusting if, in your opinion, it is morally wrong. For example if somebody behaves in an extremely selfish way, such as refusing to help a family member in trouble, you could say 'I thought John's behaviour was disgusting'. This 'disgusts' you in a moral sense, rather than physical, so you wouldn't use 'gross' in this situation.