First, we should acknowledge that neither Britons nor Americans agreed amongst themselves about politics, including the nature of the proper relationship between the colonists and Great Britain. Some members of Parliament, and other public and political British figures, openly supported American protests against the British, at least until the Declaration of Independence. Likewise, many Americans believed in remaining loyal to the Crown, and among these Loyalists there were both reform-minded liberals and conservative hardliners.
To the extent we can generalize, mainstream public opinion in Great Britain and the colonies would have agreed on the principles of limited, representative government. They generally approved of the concept of a constitutional monarch, and they agreed that individuals had certain rights, including protection from arbitrary arrest and imprisonment, protection from arbitrary taxation, the right to petition legislatures, and so on. The biggest differences between the colonists and the rulers of Great Britain had to do with how these rights and privileges—most famously the protection against taxation without representation—applied to the colonies.
The colonists claimed that they, as British subjects, could only be taxed by their own assemblies, where they were represented. Parliament, on the other hand, asserted the right to legislate for the colonies, including the right to tax. The imperial crisis revealed other differences as well, relating to the sanctity of the right to a jury trial and the right to regulate such international commerce as the slave trade. So, in short, the colonists, or at least those who became revolutionaries, believed that their rights as British subjects were identical to those in Britain itself (however, they also believed that Parliament could not fully represent them). The ideological differences between these two groups were not all that fundamental on paper, but in practice, they became irreconcilable.