Turbulence is a much broader concept, it implies the existence of vorticity and some other features, while vorticity alone does not imply the existence of turbulence. I quote:
Every aspect of turbulence is controversial. Even the definition of
fluid turbulence is a subject of disagreement. However, nearly
everyone would agree with some elements of the following description:
(1.) Turbulence is associated with vorticity. In any case, the
existence of vorticity is surely a prerequisite for turbulence in the
sense that irrotational flow is smooth and steady to the extent that
the boundary conditions permit.4
(2.) Turbulent flow has a very
complex structure, involving a broad range of spaceand time-scales.
(3.) Turbulent flow fields exhibit a high degree of apparent
randomness and disorder. However, close inspection often reveals the
presence of orderly embedded flow structures (sometimes called
(4.) Turbulent flows are three-dimensional
(unless constrained to be two-dimensional by strong rotation or
stratification), and have a high rate of viscous energy dissipation.
(5.) Advected tracers are rapidly mixed by turbulent flow.
(6.) Turbulent flow fields often exhibit high levels of intermittency.
(Roughly speaking, a flow is intermittent if its variability is
dominated by infrequent large events.)