What major disagreement about politics did Americans and British have in the years 1763 to 1775?

The disagreement between America and Britain was over
Britain's perceived denial of the "rights of Englishmen" to the Americans, particularly
the right to be taxed by their duly elected representatives. The colonies had exercised
a substantial degree of self government. Colonial legislatures imposed local laws
(subject to approval by the royal governor) and enacted laws affecting the local
populace. Even the royal governor was not wholly beyond the control of the colonial
legislatures, as they paid his salary. If he were not cooperative, they wouldn't
authorize his salary, so a workable arrangement usually was had. So self-government was
not the issue.

The problem arose immediately after the
Treaty of Paris of 1763 when Parliament decided that the colonies should be responsible
for part of the cost of the war. The war had been fought in the Americas and by British
soldiers who were paid by British funds; it only seemed proper to George Grenville, then
Prime Minister, that the colonies should pay some part of the cost of the war. The tax
imposed on the colonies was the famous Stamp Tax, which was at a much less rate than the
British population had been asked to pay, and in fact never raised any
revenue--opposition to the tax was so great that Parliament then withdrew it. The
colonists argued that as the King's good and loyal servants (which they considered
themselves) they were entitled to be taxed only by their duly elected representatives,
namely the colonial legislatures. Parliament countered with a number of responses,
including the idea of "virtual representation; that is that the members of Parliament
represented the entire British Empire including the colonies so they had "virtual
representation" in Parliament. Needless to say, the colonists did not buy this
idea.

Even after fighting broke out, the colonists offered
to mend fences if only they were guaranteed their rights as Englishmen, as stated in the
Olive Branch Petition. George III refused to even read the petition; and the war
continued to its ultimate conclusion.

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The apprehension of being degraded into a state
of servitude from the preeminent rank of English freemen, while our minds retain the
strongest love of liberty, and clearly foresee the miseries preparing for us and our
posterity, excites emotions in our breasts which, though we can not describe, we should
not wish to conceal. Feeling as men, and thinking as subjects, in the manner we do,
silence would be disloyalty. By giving this faithful information, we do all in our power
to promote the great objects of your royal cares, the tranquility of your government and
the welfare of your people.

We ask but for peace, liberty,
and safety. We wish not a diminution of the prerogative, nor do we solicit the grant of
any new right in our favor. Your royal authority over us, and our connection with Great
Britain, we shall always carefully and zealously endeavor to support and
maintain.