Because we're awesome.

England, Scotland, Wales etc aren't really countries per se, they're more like US states but with a stronger sense of identity really.
Britain = England, Wales, Scotland.
Great Britain = England, Wales, Scotland, Northern Ireland.
UK/United Kingdom = England, Wales, Scotland, Northern Ireland...and maybe all the other territories we own too, like Gibraltar and Falkland Islands.

It may be wrong, but that's how I like to think of it