Earlier quoted context omitted.
MM/DD/YYYY is clearly worse. Not only is it only used in a tiny number of places, making it more likely to cause confusion, but it doesn’t go consistently from smaller unit to bigger unit or vice-versa. YYYY-MM-DD is ideal, of course, due to easy sorting and no ambiguity over the order of MM and DD, but I’d take DD/MM/YYYY over MM/DD/YYYY any day.
+1 on ISO 8601, but MM/DD/YYYY does have a certain logic to it. The month essentially tells you what time of year it is. "Want to go on vacation in July 10th?" carries less mental overhead than "..on the 10th of July", unless it's currently July (ie, you just say "..on the 10th"). Because "in July" is more meaningful than "on the 10th of any random month".
I could make similarly silly arguments in favour of dd/mm/yy - if you're writing it down on a paper list and dates aren't yet final, it's easier to cross out the first item as you'll have space in the margins to put in the new value rather than the middle item, and for that kind of short term projections, the day is more likely to change.
The real reason we find a system preferable is just familiarity, no matter what post-facto justifications people try bringing up.