Georgia not joining the American Revolution would probably have about as much impact on the war itself as the Floridas not joining the Revolution. Like Sucrose said, it was a peripheral colony, and we'd learn about the Twelve Colonies in history books instead of the Thirteen.
Where things get interesting is
after the Revolution. Britain would be in a much stronger position to hang onto the Floridas with Georgia still on their side.
Georgia, the Floridas, and the Bahamas together could form the core of perhaps a second British dominion in North America, meaning a second destination for British loyalists (especially those from the southern colonies); Canada is likely to be more French and more slowly settled than IOTL. In addition, America is now flanked to its north and south by British territory, which will have repercussions in the next war between them.
In addition, there's the issue of the Five Civilized Tribes. They all lived in
territory that will likely be claimed by the British at the *Treaty of Paris. This butterflies away the Trail of Tears, given the better relations that the British had with the Indians and the fact that Andrew Jackson won't be running the show down there. Would they fare a bit better than IOTL? Maybe get their own territory or buffer state in OTL's northern Mississippi and Alabama?
Lastly, what happens to abolitionism in Britain, and
how do Georgia, et al. react to it?
EDIT: This thread reminded me of Glen's
Dominion of Southern America TL.