The distinction between brier and briar is nil when the word is familiarly applied to spiny plants.

Digression-
The OED has this for the Erica whose root is used to make pipes:
Brier, briar, formerly bruyer, heath; erroneously identified with the preceding word [the brier-briar of the song]. The white heath, Erica arborea.
In 1869, in the Tobacco Trade Review, the word briar, later brier, began to be substituted for bruyer.
Why bruyer was dropped is not explained; perhaps it was hard to wrap the tongue around. At the time Meerschaum (a corruption by the Germans of the translated Persian name kef-i-darya for a form of sepiolite- sea foam) was a sharp competitor.