The term _spectral invariant_ refers to an object, such as a function $Z$ of one real variable, defined in terms of a Riemannian manifold $(M, g)$ but that depends only on the spectrum of the $g$-Laplacian on the space of square-integrable functions on $M$.
Isometric manifolds obviously have equal spectral invariants, but the converse is not _a priori_ apparent (and in fact not true, hence the concept of _isospectral_ manifolds and the field of _spectral_ geometry).