Many American commentators have noted differences in the use of these two words. The usual observation is that meantime normally functions as a noun ("In the meantime,...") and meanwhile as an adverb ("Meanwhile,..."). The use of meantime as an adverb and meanwhile as a noun is generally discouraged, although most commentators allow that such usage is not incorrect; one of them (Bernstein 1971) defends the adverb meantime at some length, noting its frequent use by Shakespeare.
Meantime I've had a letter from Paulhan and have written him —Archibald MacLeish, letter, 8 July 1949
Meantime there was a core of older contributors — Times Literary Supp., 19 Feb. 1971
Meantime, he is headed in the right direction —E. B. White, letter, 4 Feb. 1974
Meantime, ... those now in American racing have successfully cultivated a reserved, dignified image — Clive Gammon, Sports Illustrated, 1 Dec. 1986
And in the meanwhile, mum's the word —Alexander Woollcott, letter, 19 Nov. 1934
... were being developed in the meanwhile by engineers —S. I. Hayakawa, ETC, Summer 1952
But in the meanwhile a lot of people learned to read —Bergen Evans, "The Language We Speak," speech, June 1968
... gained a lot of other things in the meanwhile — People, 27 Apr. 1981