The declaration of independence declared the colonies independent from Britain, so why were we fighting the British until the treaty of Paris was signed? Also, what is the difference between the declaration of independence and the treaty of Paris?
The answer to your first question is really very simple: the British government of Lord North did not recognize American independence and fought a war to keep the colonies in the empire. Fighting actually began more than a year before the American colonies formally declared independence; why would the British, already engaged in a major war, stop just because the Americans said they were independent?
As regards your second question, the Treaty of Paris was a formal diplomatic treaty which ended the war and settled a number of outstanding issues between the United States and Britain. The Declaration of Independence was a political document issued by the Continental Congress, in which that government declared itself to be independent and sovereign.