In the Grothendieck ring of varieties, there are ways of distinguishing classes of varieties, for example $\ell$-adic cohomology. The Grothendieck ring of stacks is a localization of the Grothendieck ring of varieties, more precisely it is obtained by inverting $\mathbb{L}$ and the cyclotomic polynomials in $\mathbb{L}$.

If I am not mistaken, to check that $\{X\}\neq\{Y\}$ in $K_0(Var_K)$, where $X$ and $Y$ are varieties (assume smooth, but not proper) it suffices to check that $\sum (-1)^i[H^i(X_{K^{alg}},\mathbb{Q}_l)]$ is not isomorphic to $\sum (-1)^i[H^i(Y_{K^{alg}},\mathbb{Q}_l)]$. Now say that $\{X\}=\{Y\}$ in $K_0(Stacks_K)$. Even when $X$ and $Y$ are actually schemes, then $p(\mathbb{L})(\{X\}-\{Y\})=0$, where $p$ is a product of cyclotomic polynomials and powers of $\mathbb{L}$, and I don't think we can use $\ell$-adic cohomology now. More generally, what if $X$ and $Y$ are actually stacks and not just schemes?

As a more concrete question (I am not necessarily interested in this, if you have a better example), what if $X$ and $Y$ are tori of the same rank, given by an explicit action on lattices?