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.
The differences aren't not very meaningful to music, but very prominent in musical pedagogy (unfortunately). In the Major (de:Dur, hart, lat: durus, en:hard) "gender", there are 7 fixed pitches, whereas in Minor (de:Moll, weich, lat: mollus, en:soft) two of them come in two flavours: the 6th and 7th can be minor or major. All of these combinations are in use, but there are some stereotypes, that lead to preferred sequences. So, for instance, in rising scales the major flavours are more often used than the minor ones. The natural Minor scale isn't very convincing by itself because its material is undistinguishable from Major, if there are no other clues.
You should know these three kinds of Minor, for examination purposes. But it's maybe more important to be aware, that there are no compositions at all in either of them.
If you look at Bach - who had surely a well-established feeling for Major and Minor - you'll see mostly only scale parts in use. It's ends or turning points are often the key dissonance for the Minor tonality: the diminished sixth. In the following example, I picked from Wikipedia, you see a scale part falling to the Major 7th, an upward jump by the diminished 6th, followed by another scale part:
So the sizes of the 6th and 7th in Minor are not controlled by one of "the three scales of Minor", but instead by the tonal (or musical) context (or purpose).
All of the minor scales you mentioned have something different. Let's take the C minor scale for example:
So, you could use any minor you like, depending on what you like.
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.