The General and the King each had definite ideas, goals and responsibilities.
Earlier Georges
George I became King of England following the death of the last Stuart monarch, Queen Anne in 1714. With no direct line of succession, the Hanoverian Princes/Electors had a somewhat convoluted kinship traced back a century to the Protestant children of King James I. He spoke no English, preferred his German courtiers, and returned to Hanover as often as possible. It was during his reign, mostly due to his lack of understanding (and perhaps interest) in his adoptive country, that the roles of ministers to the King became predominant.
His son, King George II was also German-born in Hanover. He also spoke German as his first language, and spent as much time as he could in Hanover. Somewhat crude and temperamental, he was nevertheless a fair and competent administrator, and enjoyed on-and-off general good will of his British subjects. It was he who came to rely on Robert Walpole as the first de facto Prime Minister.
George I had a difficult relationship with his son, who he considered a rival. George II had a disastrous relationship with his son Frederick, who he had left as a youngster to be educated in Germany. They did not see each other for fourteen years, and then, when they did, it was rancorous and distrustful, and Frederick became the “face” of the political opposition.
Happily (at least to George II), Frederick predeceased his father. The heir to the Throne of Great Britain became Frederick’s son, born and educated in England and who never visited Hanover.
George III, King
George III, the grandson of George II, was born in 1738, six years after George Washington was born in Virginia.
He received a stellar education, and learned English, German, French and Latin, politics and constitutional law, commerce and music and extensive and systematic studies in science. Plus fencing, dancing and riding.
His relationship with his grandfather was cool until the death of his father Prince Frederick. Once George became the heir apparent, Grandpa took more interest.
When his grandfather died in 1760, George became George III at only twenty-two. His uppermost thought was according to the doctrine of Divine Right of Kings. This meant that a monarch derived his authority from God alone, and no temporal authority could surpass it. Period. Rebellion or revolt by subjects was the worst of all political crimes.
George Washington, Virginian
George Washington was the fourth “George” so named in his family history, and unlikely that he was named for the monarch when he was born. Nevertheless the name “George” was a very popular one in early eighteenth century America.
His own education was a fraction of the extent of his British counterpart, however. When his father died, GW was eleven, and an expected classical education was not forthcoming.
His half-brother Lawrence took young GW under his wing at Mount Vernon, his patronymic plantation, and introduced him to his Fairfax neighbors. They were arguably the wealthiest, highest ranking and most cosmopolitan people in Virginia, and they recognized young George’s innate abilities, and helped advance his career as a surveyor, then a prestigious and honorable trade.
Those skills and experiences surveying “western” lands led to GW’s service with the Virginia militia, where he spent the better part of eight years, mostly during what was called the French and Indian War (in North America), and the Seven Years War (in Europe). Serving under British General Edward Braddock, the young GW learned more than military techniques and discipline; he learned the essence of command, and how to think like a commander.
He assiduously tried to receive a commission in the British Army, but was consistently rebuffed. When he finally resigned the Virginia militia, he was its highest ranking officer: a colonel.
The Georges Road to Revolution
King George III took his role as British Monarch very seriously. He was by and large a serious (rather than frivolous) man, and sincerely wished to be a fair, competent but firm ruler of his growing dominions. His recent war with France left the country in dire straits, however. He believed his American Colonies should pay its fair share of the war, and taxes were imposed accordingly. He believed his main role was to preserve his empire, and hand it over, intact, to his heir.
George Washington, once he inherited Mount Vernon and became a full time planter, was building his own estate, which was thriving under his astute business management. This led him to realize that the balance of trading power (stringently limited and proscribed by Great Britain) had become extremely skewed in favor of England. He had always considered himself a loyal British subject, but by 1770, began leaning toward independence for America. American needs, desires, talents and perhaps most importantly, character, had become different than that of Englishmen.
By the time “disgruntled” became rebellion on the part of Americans, and “coercion” became intransigence on the part of Great Britain, the lines had become clearer.
The Differences of the George Strategies
George Washington had a huge plate of responsibilities in creating his Continental Army: training, maintaining, instructing officers, logistic – and finding the resources to feed, shelter, arm and supply the troops. But the key to winning the Revolution was based on one overweening criteria: keep his army together. If it fell apart – for any reason, independence was lost.
Thus despite more losses than victories, GW managed by strategic withdrawals and retreats to keep his army alive to fight another day. That strategy was tantamount to his sub-commanders as well.
George III had his own strategy. No independence. Period.
But his Army and Navy were being bled dry financially trying to suppress a rebellion 3000 miles away. Parliament was becoming shakier in dealing with the King’s resolve.
Eventually, the need for independence had become more important to the Americans than the need to deny it by the British.
Sources:
Beschloss, Michael – Presidential Courage: Brave Leaders and How They Changed America (1789-1989) – Simon and Schuster, 2007
Chernow, Ron – Washington: A Life – Penguin Press – 2010
Randall, Willard Sterne – George Washington – Galahad Books, 2000
https://www.battlefields.org/learn/biographies/king-george-iii
Presidential History Blog 
Two very interesting men!