On the relation for the Myhill-Nerode theorem/syntactic monoid of a language
The syntactic monoid of a language $L$, which is $\Sigma^{\ast}$ quotiented by $\sim_{L}$, is usually bigger than the set of equivalence classes of $\Sigma^{\ast}$ quotiented by $\equiv_{L}$. Informally, the Myhill-Nerode relation $\equiv_L$ only cares about the prefixes of a word $w$ (since it reflects the processing of $w$ by a DFA), while the syntactic monoid has to encode the information about all possible infixes of $w$ (otherwise you won't get the algebraic structure $[x]\cdot[y]=[x \cdot y]$)