Why were England and its North American colonies unable to resolve their differences peacefully?
I think that there are two ways you can look at it.
First, you can simply say that it was time for them to split. You can say that the colonies had grown too big and developed to be willing to live as colonies any longer. Because of this, there was really no way for them to accept being ruled any more.
But Britain kept Canada, for example, as part of its territory long after the US. This brings up the second reason.
At this point, the British and Americans were sticking too firmly to their positions. Instead of compromising and doing give and take, they dug their heels in and kept antagonizing each other. Later on, the British learned their lessons and gave Canada much more autonomy. At the time of the American Revolution, they were not willing to do that because they thought it would weaken them.
So both sides became somewhat radical. The Americans thought it was okay to destroy huge amounts of private property (Boston Tea Party). The British thought it was important to assert their authority even when they were backing down (Declaratory Act). By doing these things, they drove wedges between them and made the cracks too large to be mended.
As the warden in Cool Hand Luke would say, what we've got here is, a failure to communicate. The colonies were established in 1607, and here we were, almost a century and a half later, and much of the colonies had grown into its own society. We had our own dialects. We had a mix of ethnicities and religions. We had a dual economy north and south, with a mostly independent one in the West.
So by the mid 18th century, it was difficult for us to see eye to eye on much of anything. Still, most Americans did not want to declare independence. It seemed silly. The series of events, though, through the French and Indian War and the taxations and restrictions that took place afterward, it became increasingly clear that divorce was a real possibility.
As each incident, from the Stamp Act Congress/Riots to the Boston Massacre, and finally Lexington and Concord, a gulf continually widened between the King and the colonies, so much so that when cooler heads tried to prevail with the Olive Branch Petition, it was too late.
Things between the colonists and the British had reached a natural point where separation seemed inevitable. With the growing colonial resentment and anger towards the British, diplomatic and peaceable solutions had become impossible to pursue. The British perceived the colonists' actions as signs of disrespect and ingratitude. At the same time, the colonists felt that the British expressed the same sentiments towards them. Both sides felt that the other had passed a point of no return in that their relationship had been tattered and torn beyond repair. In the end, it was this feeling that the relationship was beyond being salvaged that allowed both sides to see war and military conflict as the only potential solution to the problem.
England and American colonies were unable to resolve their differences peacefully because the England was interested primarily in exploiting their colonies, including those in America for serving their own selfish interest at the cost of the people of the colony. England was not looking for a mutually beneficial solution. They were only interested in their own benefits, and believed that they can forcibly crush the resistance of people of American colonies and force them to accept an unjust and unequal deal. Under these condition it was not possible to achieve a peaceful settlement with people who were determined to assert their independence and, if required, take by force what is rightfully theirs.