There seems to be no reason to choose one over the other. OED gives:
brier | briar, n.2
The White Heath ( Erica arborea), a native of the south of France, Corsica, etc., the root of which is extensively used for making tobacco-pipes (introduced into England about 1859); also a pipe of this wood.
4. briar - a pipe made from the root (briarroot) of the tree heath
briar pipe
pipe, tobacco pipe - a tube with a small bowl at one end; used for smoking tobacco