I guess the cultural significance likely isn’t too massive and I suppose that at first glance the date might not even stand out. I would put money though that more Brits know what happened on the 4th of July though than say the 16th of August. As I mentioned in another comment the events of American Independence happened far earlier than any of the other colonies and under massively different circumstances which were much less voluntary on the British side. How many of the other colonies successfully gained independence through a war with the British empire at the peak of its power rather than due to treaties passed in an era with massive public sentiment against imperialism.
That makes sense. Thanks!