Political linguistics. Portraying the war as a 'revolutionary' effort painted the rebels in a heroic light: they were overthrowing the imperialist establishment. This also appealed to a number of people living in Britain, mind; it became lucrative for businesses to sell accessories that referred to the 'revolutionary' cause. (Anything to bash the government with.)

However, journalists and government documentation at the time using 'war of independence' at least suggested an alternative to the demonisation of the British regime. If the Americans were deemed to have little to 'revolt' against, which the term 'revolutionary' refers to, a different description suggested their motivation was wanting to become an independent state more than making a stand.

(I will say that, as a Brit myself, it usually is referred to as the American revolution here, at least colloquially. If anything, 'war of independence' is more an academic term in modernity.)