I'm studying scales and came across the natural, harmonic, and melodic minors. I understand the patterns for each scale but what other important differences exist between the three?
There are at least some differences in how they're used.
In short, you start with the natural minor scale. You use the harmonic minor scale when you need a dominant chord, and modify it to be the melodic minor if you need the leading note in the melody. This all is of course simplified but should give the basic idea.
I'm going to expand upon the answer already provided just to give some further clarification on the history and usage of each scale.
C Major = C D E F G A B C
This is true for all natural minor scales.
As the scale ascends, the 6th and 7th degrees are raised in order to dissolve the augmented interval between the normally b6 and #7 scale degrees and create a leading tone that resolves strongly to a tonic. Thus:
A Melodic Minor = A B C D E F# G# A (ascending)
As the scale descends, it becomes a natural minor scale, with the 6th and 7th degrees now moving back to their original formation.
A Melodic Minor = A B C D E F G A (descending)
Hope that helps.