What is the difference?
Astronomy, both observational and theoretical (and now also experimental!) is about what is going on in the universe that we can see now, or extrapolate to short times (~a billion years) in the past or future.
Cosmology, is about how the universe developed, and how we can infer that from the data provided by astronomy.
Observational Cosmology = Astronomy on the largest possible scale.
"On the inside looking out"
Theoretical Cosmology = Modelling the universe as a whole.
"On the outside looking in"
Separate names with a comma.