What is the difference between "Hydrophobia" and "Aquaphobia"?

There is no "fundamental" difference between aquaphobia and hydrophobia - both mean (irrational) fear of water - except that in the first, the two root words are Latin and Greek (respectively) in origin, and in the second, both root words are from Greek.

That does not mean, though, that they are interchangeable. Historical use differentiates between the two.

Hydrophobia is the only term used for the phenomenon of pharyngeal spasms occurring in mammals infected with the rabies virus. The difficulty swallowing results in drooling of excess saliva instead of swallowing, and an aversion to liquids, which was labeled (not too accurately) "hydrophobia".

Aquaphobia is an irrational fear of water, specifically fear of potential consequences of entering the water (i.e., drowning).

A doctor would never use the word aquaphobia to describe a symptom of rabies, nor would a careful (?pedantic?) psychologist use hydrophobia for an irrational fear of water (the DSM uses hydrophobia as a symptom of rabies.)

Also, hydrophobic is a fixed term in Chemistry.