I have been dancing for a couple of years now, and I notice different places start the box step differently. Some start forward with the left foot on a slow(as a lead) and some places start side together with the left foot with quick, quick. What is the main difference of the two? Just curious, thanks.

I am not sure if there is a preference related to American vs. International style, but I have been taught both ways as well. My personal preference is the side-together with the left foot start (QQ). For some reason, it makes it easier for me to go into a crossover or a fifth position break sequence or even a cross-body lead or Cuban Walk. Couldn't tell you why that is, other than it's a "comfort zone" thing.
Good question, though. I have often wondered myself.
I want to say Terence or Jonathan addressed this a couple of years ago, but not sure.