Clerical bands are a relic of the ancient amice, a
square linen tippet tied about the neck of priests during the
administration of mass (Discontinued by the parochial clergy the latter
part of the 19th century, but still used by clerics on the Continent.)
Legal bands
are a relic of the wide collars which formed a part of the ordinary
dress in the reign of Henry VIII, and which were especially conspicuous
in the reign of the Stuarts. In the showy days of Charles II the plain
bands were changed for lace ends.
The eighth Henry, as I understand,
Was the first prince that ever wore a band.
John Taylor, the Water Poet
(1580–1654).
Source: Dictionary of Phrase and Fable, E. Cobham Brewer, 1894