I'm not sure if this question is appropriate for this subforum because it reffers to b-school application generally, not only to part-time programs. Nevertheless, question is sound and certainly interesting for many candidates.
From my own experience, applying early doesn't make any advantage. Lets assume two extremes, of course - we are talking of borderline candidates overher, people with good GMATs, GPAs but also with some weaknesses in profiles.
First - a candidate from underrepresented country or profession. If you are the only applicant from your country who submitted the application in R1, there is great possibility that the adcom would put you on the waitlist and wait for someone else from your region and/or profession to apply. In R2 you'll be compared to interviewed applicants and probably notified of the deceison. If nobody else with similar statistics applied, or applied but not as good as you are, you will be probably offered the place.
Second - a candidate from overrepresented country or industry. If you are one of these, there is even greater probability you'll be WLed in the first round. Simply, adcoms know there are plenty of similar applicants and they want to pick the best ones. Consequently, they are keen on waiting untill mid January, admitting the best candidates and waitlisting the others. Again, they are going to consider all the waitlisters against R2 applicants.
I really don't see the difference between two rounds, even slightest. The only difference might be the quality of your application. The best advice is to apply when you are ready. If you are ready for R1 - go for it. If you're not, you have time untill the first week in January and not a minute more. You have to apply in R1 or R2, especially if you are international applicant.
My GMAT experience
My retake experience - GMAT dropped from 720 to 690 - advice for retakers
My TOEFL experience - 108