The main difference between the two I see is that partial application basically binds a certain argument to a fixed value, while currying only changes the way the function is called (thus allowing for easier partial application). It is easier to see on a function with more than two arguments. Imagine f:(A×B×C)->D. Currying it will yield f':A->(B->(C->D)), so you would call it as f'(1)(2)(3). On the other hand, partially applying on first argument it would yield f'':(B×C)->D.
The main difference between the two I see is that partial application basically binds a certain argument to a fixed value, while currying only changes the way the function is called (thus allowing for easier partial application). It is easier to see on a function with more than two arguments. Imagine f:(A×B×C)->D. Currying it will yield f':A->(B->(C->D)), so you would call it as f'(1)(2)(3). On the other hand, partially applying on first argument it would yield f'':(B×C)->D.