Hi Swandiver. I think your question is a good one. To me an arpeggio is a pattern generated using notes that are fingered and octave transpositions thereof. A style can play an arpeggio, but is not limited to notes fingered.
Of course, one can program a style to arp. So I have been trying to think what can one do with an arpeggiator that one cannot do with styles (at least as they are implemented today). One difference would be that you can select your octave with an arpeggiator, but not with a style. Another difference is that you can arpeggiate any combination of notes, whereas there might be some note combinations that would not be recognized by a style player. A third difference is that you can arpeggiate two and three-note combinations, but you get either one or the other with a style. A fourth difference is that you can select between open and closed voicings while playing an arpeggio, whereas the voicing is usually fixed with a style.
In summary, while you can use a style to arpeggiate, you have more control over arpeggios when you use an arpeggiator instead of styles.