How did the British and the colonists differ on the issue of taxes?

I assume you are asking about the difference between the colonists and the British government.  If so, the basic difference is that the British government believed that it had the right to tax the colonists under almost any circumstances.  By contrast, the colonists felt that the British government had the right to tax them only in very limited circumstances (and perhaps not at all).

The American colonists felt that the British government only had the right to tax them in very limited circumstances.  They (of course, this is just a generalization as not everyone felt this way) believed the government had the right to tax imports and exports which were things that were external to the colonies.  However, they believed the government did not have the right to tax them on internal things (like the Stamp Tax did).  Some colonists even believed that the British government did not have the right to tax them at all because they were not represented in Parliament.

By contrast, the British government claimed the right to tax the colonists in all circumstances.  The government believed that, because the colonies were part of the United Kingdom, it had the right to tax them even on internal goods.  The British government believed that the colonists were represented in Parliament through the idea of “virtual representation.”  This idea held that the members of Parliament would represent the colonists because they represented everyone in the empire. 

In these ways, the colonists (or at least many of the colonists) and the British government had very different ideas about taxation.