In ordinary category theory, coequalisers are always epimorphisms, so effective epimorphisms are indeed epimorphisms. But this is not so in higher category theory.
Indeed, consider the unique morphism from the circle S1 to the point Δ0. This is surjective on connected components, so it is an effective epimorphism in the (∞, 1)-category of spaces. But it is not an epimorphism: indeed, a morphism X → Δ0 is an epimorphism in the (∞, 1)-category of spaces if and only if the (unreduced) suspension of X is (weakly) contractible, and the suspension of S1 is the sphere S2, which is certainly not contractible.
On the other hand, there are non-trivial epimorphisms X → Δ0 is the (∞, 1)-category of spaces, i.e. there is a space X whose (unreduced) suspension is (weakly) contractible: see here for details.
[Thanks to Aaron Mazel-Gee and Saul Glasman for helping me figure this out.]