The motives of the colonies and the British dovetailed because they both wanted to defeat France. From there, the interests of the mother country and colonies diverged.
The British were interested in defeating France worldwide. The French and Indian War was merely one small part of the overall Seven Years' War. The British were going to do whatever was necessary to secure their overall empire rather than doing things that would be explicitly meant to advance the colonies' interests. For example, one reason the British wanted the French out of America was the fact that the French were trading too much with the colonies, bypassing Britain and defeating the mercantilist point of having colonies.
By contrast, the colonies simply wanted France out so that the colonists could expand into areas controlled by France. The British did not really care about this. In fact, after the war, they issued the Proclamation of 1763 to prevent this.
The colonies wanted the war to be waged for their own benefit while the British wanted to wage the war as part of an overall plan to defeat France. Because of this, their motives diverged a great deal.
Studying the French and Indian War completely from the Native American perspective, explain the actions of the major Indian federations and explain why they often seemed to be on the winning side of the battles but still came to what we now know is a tragic ending.