Conjugation vs Change of Basis

For transformations, A and B are similar if A = S^-1BS where S is the change of basis matrix.

For Lie groups, the adjoint representation Ad_g(b) = gbg^-1, describes a group action on itself.

The expressions have similar form except for the order of the inverses. Is there there any connection between the two or are they entirely different concepts?

A and B are also similar if there exists an invertible [itex]S[/itex] such that [itex]A = SBS^{-1}[/itex].