Methods of Weighted Residuals (MWR)  usually include Galerkin, collocation, method of moments, least-squares and subdomain methods.
Spectral methods  usually include Galerkin, tau and pseudospectral methods. Pseudospectral is another name for collocation and the tau method is the same as the method of moments .
I can't see a difference. If there is no difference between them, why the different names? I'm writing a monograph about these methods and want to make sure I'm not missing something. If you'd like to see my unfinished monograph, it is at http://tildentechnologies.com/Numerics
 Young, L.C., “Orthogonal Collocation Revisited,” Comp. Methods in Appl. Mech. and Engr. 345 (1) 1033-1076 (Mar. 2019) (doi: 10.1016/j.cma.2018.10.019).
I tried this over at the Mathematics SE but got no response.