GB is not "more neutral".

Great Britain is the name of the post 1707 Union of England and Scotland. It's not neutral if you are a member of the Unionist community in Northern Ireland.

The United Kingdom is the name used by the country at the UN but is, of course, not neutral if one is a member of the Nationalist community in Northern Ireland.

No one uses .co.gb domains.

Here .... en-ie and en-gb are different things, because the Irish English dialect is a bit more distant from British English than the Scottish one. Whereas .uk is a country code representing a specific political entity.

GB is not more neutral - it excludes Northern Ireland, which is a part of the UK.

Aside from any such associations, they mean different things.

People often use the terms imprecisely, but (for example) if Scotland were to leave the United Kingdom, it'd still be part of the island of Great Britain.

It doesn't seem very neutral to go around demanding others call you great.

Yes I know.. that's not how the word is meant in this context...