It is called the Union Flag because it symbolises the administrative union of the countries of the United Kingdom. It is made up up of the individual Flags of three of the Kingdom's countries all united under one Sovereign - the countries of 'England, of 'Scotland' and of 'Northern Ireland' (since 1921 only Northern Ireland has been part of the United Kingdom). As Wales was not a Kingdom but a Principality it could not be included on the flag.
Great Britain is the official name given to the two kingdoms of England and Scotland, and the principality of Wales. The United Kingdom (UK) includes these areas as well as Northern Ireland. Britain refers to only England and Wales, though some will shorten Great Britain with this term.
This royal throne of kings, this sceptred isle, This earth of majesty, this seat of Mars, This other Eden, demi-paradise, This fortress built by Nature for herself Against infection and the hand of war, This happy breed of men, this little world, This precious stone set in the silver sea, Which serves it in the office of a wall Or as a moat defensive to a house, Against the envy of less happier lands,-- This blessed plot, this earth, this realm, this England.
Correction, flag of the United Kingdom. The flag of Great Britain doesn't have the St Patrick cross as Ireland was not fully incorporated under the crown until 1807 I.e the formation of the United Kingdom of Great Britain and Ireland. Wales is not featured because it's a principality, not a kingdom.