As for New France, the Abenakis and their allies were effective in terrorizing the Thirteen Colonies to a point where the American Rangers made a great effort to burn down Odanak and slaughter its population in the middle of the winter of 1759. Multiple raids occurred before that date, such as the ones in Deerfield or Wales. Native Americans were more effective at asymmetrical warfare, though. As for conventional battles with two modern armies facing each other, their lack of firepower and command structure was a big disadvantage. For example, in 1666, the French sent part of the 2400 men strong Regiment of Carignan Salières in the Iroquois country, burning down the villages and slaughtering a few children and elders without significant opposition. Historian Richard White's classic "Middle ground" hypothesis states that Native Americans mostly managed to pit European powers against one another in order to keep their autonomy. When France was gone, this balance tipped in the British favor.

As far as psychological warfare goes, they had a tremendous impact on European conceptions of social life, exerting the fascination of waves after waves of missionaries, scientists, anthropologists... Some even argued that the Huron took revenge on the French by showing them a life without a King, an idea that would ultimately lead to the Revolution. This thesis is, I agree, a bit far-fetched, but still it manages to illustrate the sheer power of winning the minds. Same goes with the contemporary revival of Native American culture, which makes taking military action against Native Americans in the XXIst Century almost unthinkable, or at least very very unpopular. They might have been outmatched on the battlefield, but recent experiences in Afghanistan, Irak or Vietnam can teach us that you can very well lose a war by winning all the battles.

Although I am not a specialist of that area, the Aztecs were known to have a more significant organization and fighting ability against the XVIth Century Conquistadors, to a point where Cortes famously evoked the idea of joining forces with Moctezuma in order to conquer China. Still the Aztecs were defeated, mostly by disease and internal fighting. As with the Huron, the actions of the Conquistadors would bring infamy on the Spanish Empire, giving birth to a "black legend" that would be an important part of the contestation of Spanish dominance in Europe.