Criterion of nonsingular varieties

Question

It's well-known fact, that if $X$ is non-singular algebraic variety over algebraically closed field $k$ and $Y \subset X$ is its irreducible closed subscheme defined by sheaf of ideals $J$, then $Y$ is non-singular iff $\Omega_{Y/k}^1$ is locally free and the sequence $$0 \to J/J^2 \to \Omega_{X/k}^1\otimes \mathcal O_Y \to \Omega_{Y/k}^1 \to 0 $$ is exact. I want to find examples when one of these conditions is not sufficient (neither implies the other). I think that there should be such affine examples, but I have no idea how can I construct it.