The quick answer to your question is, no, there is no fundamental difference in weaponry used by the colonists and British during the war. Typical infantrymen in both armies used flintlock weapons (mostly smoothbore, though some rifles were used by both sides), often with a bayonet as a shock weapon. In terms of artillery, again both sides used the same types of weapons for different roles: cannons for anti-infantry, and howitzers and mortars as specialized siege weapons.

In terms of new weapons, the Ferguson rifle was on of the first attempts at the introduction of a breech-loading (rather than muzzle-loading as was common at the time) battle weapon. However, its usefulness was dubious as it was expensive to manufacture and as such was only introduced in a very small number during the war.