I can't see a difference. 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.
 Finlayson, B.A., The Method of Weighted Residuals and Variational Principles, Academic Press, New York, NY (1972), SIAM Classics in Applied Mathematics (2014).
 Canuto, C., Hussaini, M.Y., Quarteroni, A. and Zang, T.A., Spectral Methods in Fluid Dynamics, Springer-Verlag, Berlin (1988)
 Young, L.C., “Orthogonal Collocation Revisited,” Comp. Methods in Appl. Mech. and Engr. 345 (1) 1033-1076 (Mar. 2019), doi.org/10.1016/j.cma.2018.10.019,