Is there a qualitative difference between the enthalpy of a phase transition versus the enthalpy of a heating or cooling process? If so, explain those differences.

No

Explanation:

given that, enthalpy is a state function, that means it depends only on the initial and final states, there is no difference between the enthalpy of a phase transition versus the enthalpy of a heating or cooling process, when the cooling or heating process finish in a change of phase.

It does not matter which way we take to cool or heat the substances the Enthalpy of this process will be the same.