If you see the Abbots Bromley horn dance performed in the US (Revels, Pinewoods, etc) it typically looks something like:

(youtube)

Note the serious and mildly "spooky" setting, and the use of Robinson's tune.

Compare to how it's done in Abbots Bromley itself:

(youtube)

Note the happy bouncy tune and lack of theatrical mystery.

Sometimes this leads Brits to accuse Americans of taking their dance and botching it. But the real story is a bit more complex!

The history of folk traditions is often murky, but in this case there are written sources. In The Sword Dances of Northern England Together With the Horn Dance of Abbots Bromley (1911, p105-106) Cecil Sharp has:

There is no special or traditional tune for the dance. The musician told me that any country dance air would serve, provided that it was played in the proper time (in Common Time, 108 bpm). When I saw the dance performed two tunes only were played, "Yankee Doodle" and the following simple little melody: [snip]

In a letter written in 1893 by the vicar of the parish (see "Folklore Journal," vol. iv, pg. 172), it is stated that a special tune used to be played for the horn dance by a man with a fiddle and within the memory of some men living, but that all efforts to recover it had failed.

But then, in The Sword Dances of Northern England, Song and Dance Airs Book II (1912, preface) Sharp has an update for us:

The Abbotts Bromley Horn Dance Air was sent to me by Mr. J. Buckley, who noted it down in 1857 or 1858 from "the fiddling of William (or Henry) Robinson, a wheelright of Abbotts Bromley, who is famous at the time is the only man who could play the horn dance air."

This is the same Robinson's tune as used in the US today (more on the tune). Musicians in the village apparently had moved on to other tunes, as is common with folk dances.

At some point in the 20th century, the Thaxted Morris Men (Thaxted, England) put together their own version, which they would dance at the closing of the Thaxted Annual Ring Meeting. This included both the revival of Robinson's tune and the introduction of a more theatrical framing:

(youtube)

Here's a description from the November/December American Morris Newsletter (pdf):

This continued until 10:15 p.m. when everyone fell into a dead hush awaiting the Abbots Bromley Horn Dance, performed annually by Thaxted M M. This was a macabre experience, with the familiar haunting melody played on a solitary fiddle, interspersed only by the occasional, but regular, 'ding' of the triangle. The dance lasted about 15 minutes, Thaxted M.M slowly appearing and disappearing from a narrow alleyway leading to the church, where the horn are stored year-round.

Ben Cowell says Thaxted has been dancing the horn dance since 1926, though I haven't been able to corroborate. I also don't know if they've been dancing something like their current version this whole time.

In the US it's been danced since at least the first Christmas Revels in 1971:

In December, 1971, the Pinewoods Morris Men were a featured attraction at the first Christmas Revels at Sanders Theatre in Cambridge. The performance included a stick dance (Balance the Straw?), the Abbots Bromley horn dance, and a longsword dance in John Langstaff's version of the "St. George" play.

While it's possible that the Thaxted and American versions and descend independently from Sharp's description, they're more similar than I'd expect if that had happened. Instead, my best guess is that the American version traces to Pinewoods, then to Thaxted, and then via Sharp's books back to Abbots Bromley.

Whether this makes the Thaxted/American version more or less traditional than the Abbots Bromley version isn't something I'm going to get into, but the dance (and very likely setting) aren't something that Americans just made up.