A lot of people would think it'd change the world a ton and no doubt it would, but a USA just avoiding the Revolution and staying a British dominion I think would end up pretty broad-strokes similar to real life and in turn affect the world the same way Remembering the Thirteen Colonies were always autonomous and fought to stay that way, let's say a miracle happens and the Olive Branch Petition is accepted.
Now compare some broad strokes similarities in real life that happened between the independent USA and still-British Canada: they got constitutional developments (1789 Constitution and 1791's Canada Act), expanded westward from then to about the 1840s territorially, got dragged into the Napoleonic Wars in the 1810s, had governmental crisises in the 1830s (Nullification Act and Canadian Rebellions), adventurism and provincial governmental changes in the late 1840s to early 1850s (Mexican War and Crimean War, Compromise of 1850 and responsible government starting up in the Maritimes), federal supremacy and crushing natives in the 1860s (Union winning in the Civil War and Confederation, Great Sioux War/Indian Wars and Red River Rebellion), first non-North American wars in 1898 (Spanish American War and Boer War), then entering the world wars and settling into world cultural supremacy from the middle 20th century to now.
We can pretty easily extrapolate that and merge these parallels into a full Dominion of (North) America's history - Canada formally included in this Dominion BTW - with dominion status happening in the late 1780s-early 1790s or so, taking over most of North America (Louisiana, Florida, Texas, maybe even pushing west to the Pacific already) in the Napoleonic Wars with a distracted France and Spain, moving west to the Pacific by the 1840s tops and crushing the last powerful native tribes by the 1860s, some local rebellions post-Napoleon if not a huge one/Civil War over slavery and local gentry in the 1830s (combining the Empire's slavery ban in 1833, Canadian Rebellions, the Nullification Act), likely a Balfour/Westminster-style full independence in the 1840s by the time of real-life Canadian self-government and American western organization... but still helping Mother Britain out in both imperial adventures (Boers, eh) and then the world wars till it becomes the big boy amongst Anglos. A republican America already throws its weight around the world, one that still had the British king as head of state or parliamentary government certainly wouldn't be adverse to do so even if it had a 'nice', gradual split up from Britain.