In what ways is Appalachian speech closer to Elizabethan English than contemporary British?

Wikipedia mentions there could actually be 3 for Appalachian English:

• One theory is that the dialect is a remnant of Elizabethan (or Shakespearean) English that had been preserved by the region's isolation.⁽²⁾⁽³⁾
• Another theory suggests that the dialect developed out of the Scots-Irish and Anglo-Scottish border dialects brought to the region by some of its earliest British Isles settlers.⁽⁴⁾
• Recent research suggests that Appalachian English developed as a uniquely American dialect as early settlers re-adapted the English language to their unfamiliar frontier environment. This is supported by numerous similarities between the Appalachian dialect and Colonial American English.⁽⁵⁾

You can listen more about Elizabethan English in this "Voices recording" session, with the following comments:

The eminent Shakespearean scholar, John Barton, has suggested that Shakespeare's accent would have sounded to modern ears like a cross between a contemporary Irish, Yorkshire and West Country accent - and cites the present-day speech of the Appalachian Mountains as the most suitable model for actors attempting to imitate a period performance.

Traditional dialects are above all characterised by their retention of conservative forms of speech - old-fashioned vocabulary, non-standard grammatical constructions and archaic pronunciations - long after the prestige language has changed.
This is, of course, true of the dialects Barton refers to and explains why, for instance multiple negation - constructions such as "I ain't done nothing" - are perfectly acceptable in Shakespearean English and modern dialects, but not in contemporary Standard English.

However, this paper puts the so-called proximity with Elizabethan English in perspective:

Focus of numerous rumors and myths, the language of Appalachia has been alternately lauded as pure and unadulterated Elizabethan English and condemned as a lazy and ungrammatical corruption of modern American English.
Both these accounts are far from the truth—although Appalachian students cannot read Shakespeare with any greater ease than students from Texas or Vermont, Appalachian English is a full-fledged language with rules of discourse, pragmatics, phonology, and syntax.

Michael Montgomery says, “The idea that in isolated pockets somewhere in the country people still use “Elizabethan” or “Shakespearean” speech is widely held and is one of the hardier cultural beliefs or myths in the collective American psyche.” (Montgomery, 1998)

The idea arose in the late nineteenth century [...]
At one extreme it reflects nothing less than our young nation’s yearning for a stirring account of its origins, while at the other extreme the incidental fact that English colonization of North America began during the reign of Queen Elizabeth I more than four centuries ago.

Two things in particular are responsible for its continued vitality:

• its romanticism and
• its political usefulness.

Its linguistic validity is another matter entirely.
Linguists haven’t substantiated it, nor have they tried very hard to do so, since the claim of Elizabethan English is patently based on very little good evidence.
But this lack of support is a secondary, if not irrelevant, matter for those who have articulated the Shakespearean English idea in print—popular writers and an occasional academic—for over a century.
It has indisputably become a powerful cultural belief and acquired mythic status. (Montgomery, 1998)

_{(2) Michael Montgomery, "How Scotch-Irish is Your English?" The Journal of East Tennessee History vol. 67 (1995), 17-18.
(3) Cooper, Horton. "History of Avery County", Biltmore Press, (1964),
(4) David Hackett Fischer, Albion's Seed: Four British Folkways in America (New York: Oxford University Press, 1989), 653-654.
(5) Montgomery, 1002-1004.}