Busbies are smaller and ultimately Hungarian; Bearskins derived from 18th century grenadier mitres ( some made of bearskin ) and were copied from the Imperial Guard by the British Grenadier Guards after Waterloo.
Older Guard regiments such as the Royal Life Guards in Denmark using the bearskin from 1803, and the Swedish Svea Life Guards ( unknown ) seem to precede the British and maybe the French — I don’t know if the Consular Guard that Bonaparte stole ( as he did everything ) to make his own personal guard had bearskins, although I vaguely recall a print of Lucien Bonaparte dispersing the Ancients with the soldiers wearing them.
Of course line regiments both Cav and Inf of various nations wore them in the 7 Years War, with the Austrians seeming the first, perhaps again from Hungary.
American bands seem to have copied it from the British, and of the U.S. Marine Band only the Drum Major wears one. Since the first Drum Major was appointed in 1861, when they substituted the office for that of Leader, I would guess that maybe when they adopted this cruel habit ( pun intended ).
Maybe not: Francis M. Scala became the director of the United States Marine Band in 1855. He was the first leader of the band as his predecessors had been fife and drum majors. Scala revamped the entire band including the instrumentation and the drum major roles. The key to the use of the bearskin was the fact that he patterned the band after European bands of the period including that of the British bands. The Military Academy band followed that lead but initially had the Belgic (Waterloo) Shako made of black felt. Later, photographs from 1888 of the Gilmore Band and the 1900 Military Academy Band also show visibly that the bearskin had become standard dress for drum majors.