Too big to fit well as comment: There is a seeming-technicality which is important to not overlook, the question of whether a symmetric operator is "essentially self-adjoint" or not. As I discovered only embarrasingly belatedly, this "essential self-adjointness" has a very precise meaning, namely, that the given symmetric operator has a unique self-adjoint extension, which then is necessarily given by its (graph-) closure. In many natural situations, Laplacians and such are essentially self-adjoint. But with any boundary conditions, this tends not to be the case, exactly as in the simplest Sturm-Liouville problems on finite intervals, not even getting to the Weyl-Kodaira-Titchmarsh complications.
Gerd Grubb's relatively recent book on "Distributions and operators" discusses such stuff.
The broader notion of Friedrichs' canonical self-adjoint extension of a symmetric (edit! :) semi-bounded operator is very useful here. At the same time, for symmetric operators that are not essentially self-adjoint, the case of $\Delta$ on $[a,b]$ with varying boundary conditions (to ensure symmetric-ness) shows that there is a continuum of mutually incomparable self-adjoint extensions.
Thus, on $[0,2\pi]$, the Dirichlet boundary conditions give $\sin nx/2$ for integer $n$ as orthonormal basis, while the boundary conditions that values and first derivatives match at endpoints give the "usual" Fourier series, in effect on a circle, by connecting the endpoints.
This most-trivial example already shows that the spectrum, even in the happy-simple discrete case, is different depending on boundary conditions.