I have 10+ DUE R3 models, 6 DUE R3-E models, 2 SainSmart brown-PCB DUE boards, and 1 SainSmart red-PCB DUE. All of the R3 boards and the red SS DUE fail to start up after a powerup and require a manual reset. All of the R3-E boards and the brown SS boards start up perfectly upon powerup every time. Seems like the red SS is a clone of R3 and the browns are of R3-E.
I am going to flash the 16U2 on one of the R3 boards with the reset code from this thread:
http://forum.arduino.cc/index.php?topic=167492.msg1373055#msg1373055
Meanwhile, the question to Arduino remains unanswered: "WHAT IS THE DIFFERENCE BETWEEN R3 and R3-E??"
I suspect that it is a 16u2 firmware change similar, if not identical to the one in the mentioned thread.
UPDATE: Flashing the 16u2 with the reset code from the above forum posts works. This is better solution than hardware mods.