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.

Its all to do with "clave". if dancing it QQ "side" the rhythm falls in line with that ( the 2nd bar on "2" is ON clave,, the 6th beat ).

When mambo first was introduced, it was taught breaking on "4", which is the beat preceeding the conga slap on " and", which is a clave beat.. but.. the change to the backward position of breaking on " 2 " falls more in line with the actual clave , hence the reason for the side action which was translated (QQ ) into the back break of Mambo.

Its all to do with the " call and response " from Cuban Rumba ( folk style ) .

One also has to be careful because the the clave may change from 2/3 to 3/2 without notice .

The 2 most dominant Clave rhythms in use are the Guaguanco in 2/3
and Montuno a rev Clave in 3/2.

most salsa is written in the Guaguanco style and reversals to Montuno in 3/2 is a more "folksie " side of latin rhythms.

In addition there is Guaracha in 3/2 which is the faster form of Danzon/ Rumba .

All of these come under the " Son " banner . ( Guajira is the Cha cha form )..

hope I didnt confuse you !!

PS.. this may be of interest.. The 2 chain schools have a different approach to Rumba Amer. style F/A commence side QQ.. A/M Fwd S.

The national D.D. for F/A was John Monte ( I worked in NYC with him ) and his brother was Hugo Montenegro, a world class musician.. that may have some bearing on their interpretation of Rumba.

Starting with left foot forward with SLOW is just a short cut. Starting with QUICK QUICK on the side is just a longer route. After QUICK QUICK on the side, then step forward with left foot on SLOW. SAME BANANA, YES?

LatinNite