Is the hoarseness that follows the flu different than other kinds of hoarseness?

No. Hoarseness is hoarseness whether from a cold, flu, cough, reflux, vocal abuse, allergy etc... See an ENT for a fiberoptic evaluation.
Yes. Hoarseness is a abnormality in the quality or sound of the voice. Hoarseness from the flu should resolve as the inflammation goes away. It hoarseness persists beyond 4 weeks, you should seek out an evaluation of your head and neck.