Relation Brauer algebra vs. Weyl group for so, sp  ? (For GL S_n plays two roles 1) Weyl group 2) SchurWeyl duality ) Question Is there any relation between Weyl groups for orthogonal and symplectic groups vs. Brauer algebra ?
Motivation For GL the symmetric group $S_n$ plays two roles: 1) Weyl group 2) appears in Schur–Weyl duality.
Brauer algebra plays the role of S_n in Schur-Weyl duality for SO, SP  (for both groups with the only difference that wheel = N or -N).  And it is NOT (as far as I understand) the group algebra of corresponding Weyl group. However since in GL case it is so, may be there is some relation  ?
 A: I think the answer to your basic question is negative, though I don't have all the relevant literature at my fingertips.   While $S_n$ does happen to play simultaneous roles for Lie type $A_{n-1}$ as the Weyl group and as a key player in Schur-Weyl duality, this seems to be an accident.   For example, types $B,C$ (odd orthogonal and symplectic) have the same Weyl group but not the same Brauer algebras, while type $D$ (even orthogonal) has a different Weyl group.   
In terms of highest weight theory, the Weyl group always plays an essential theoretical role but this is not usually related directly to the role of a symmetric group permuting factors in a tensor power of some "natural" representation as in Schur-Weyl duality.    As far as I know, the analogy you are looking for isn't helpful.
ADDED: I should emphasize that for fixed $n$ and $\mathrm{GL}_n$ (originally over the complex field), each symmetric group $S_d$ plays a role in Schur-Weyl duality by permuting factors in the tensor product of $d$ copies of the natural $n$-dimensional module.   Only when $d=n$ does this happen to involve the Weyl group of $\mathrm{GL}_n$ (or $\mathrm{SL}_n$).   
