The Continental Congress voted 12-0 on the 2nd of July 1776 that the fledling colonies should break away and declare their independance from the United Kindgom. The official announcement and hence the date of the 4th is the date by which the terms of Jefferson's declaration should take effect.
Despite the genesis of Independence Day, it is largely uncommon for Americans to express anti-British sentiment on the day or to view it as a celebration of anti-colonialism. Indeed, most Americans today consider the United Kingdom their greatest ally. Rather than specifically as an opportunity to commemorate the end of British rule in the 18th century, contemporary Americans generally perceive the holiday as a celebration of the U.S.A. itself and the political values that motivated the United States Declaration of Independence.
Independence Day is as far as I know the only holiday celebrating the United States as a whole.
So what would have happened is those 13 colonies had not signed the Declaration? What would have been if the instrument of the United States had not existed?
I suspect that what would have happened is that America would have adopted a slightly different constituition more akin to something like Australia and been eventually awarded Commonwealth status which is different to Canada and New Zealand which were Dominions.
The Canadian Constitution which is the oldest of the 3 has the pre-amble as follows:
Whereas the Provinces of Canada, Nova Scotia, and New Brunswick have expressed their Desire to be federally united into One Dominion under the Crown of the United Kingdom of Great Britain and Ireland, with a Constitution similar in Principle to that of the United Kingdom shall form and be One Dominion under the Name of Canada; and on and after that Day those Three Provinces shall form and be One Dominion under that Name accordingly.
Unless it is otherwise expressed or implied, the Name Canada shall be taken to mean Canada as constituted under this Act.
The Constitution of Canada provides the framework by which the powers of the Provinces and the Nation are spelled out. The respective constitutions of Australia and New Zealand also act very similarly.
America instead of being called the United States of America would more than likely been afforded Commonwealth Status with with the same sort of relationship between the states and the federal parliament to Australia. And because they would be a Commonwealth would called be the Commonwealth of America, still hold a bicameral parliament and more than likely have cabinet appointed from elected members rather than the case at the moment where not one single member of the cabinet is appointed by the people.
Actually , Queen Elizabeth II in Australian, Canadian and New Zealand politics has no executive power at all, so in this regard America would probably have the same level of autonomy and sovereignty as now.
I think culturally the Commonwealth of America would be as a nation less militant than now. The rights of free citizens would have been extended to all sooner, certainly under English common law no person could be recognised as the property of another and how this came about in the USA is somewhat of a mystery to me.
Also because the constitution would be different, the 2nd Amendment which gives Americans the right to hold firearms would not exist, so I suspect that the place would be safer.
So in all consideration without the dilemmae of slavery and guns in society, America would have been a better place.