The difference is to which groups we think the assumption applies.
The discrete logarithm assumption is thought to apply to groups of the form $\mathbb{Z}_p^*$ with $p$ prime, amongst others. As explained in the linked lecture notes, there exist polynomial time algorithms that determine whether $m_b$ is a quadratic residue. This is sufficient to rule out half of the possible plaintexts, because in groups modulo a prime, exactly half of the elements are quadratic residues. Given this counterexample, the discrete logarithm assumption is consequently not sufficient to achieve semantic security.
The Decisional Diffie-Hellman assumption does not apply to groups $\mathbb{Z}_p^*$ with $p$ prime, but it is thought to apply to other groups -- the ones typically use for ElGamel. The DDH assumption is a stronger assumption that applies to some of the groups where the discrete logarithm assumption holds.
In particular, if the DDH assumption holds, we can prove that there cannot exist a polynomial time algorithm that determines whether an element is a quadratic residue. This necessarily disproves the DDH assumption for groups $\mathbb{Z}_p^*$ with $p$ prime.
So long story short, if the DDH assumption holds, then the attacker cannot perform the attack using quadratic residues in polynomial time, or in fact even distinguish chosen plaintexts in polynomial time. However, the DDH assumption does not hold for all groups where the discrete logarithm assumption holds, and for some of those groups where the DDH assumption does not hold, an attacker can perform the attack in polynomial time.