They are not related to each other but share some minor similarities in heritage (as do most of the American breeds). Both are dual purpose breeds but were created separate and independent of one another.
The RIR was created by crossing native mutts with Brown Leghorns, Cochins, Brahmas and Red Malays.
The Buckeye was created by crossing Barred Rock hens to a Buff Cochin Rooster, then a Black Breasted Game Fowl, probably of Asian descent, was top crossed over these hybrid pullets and the breed developed from there. The Buckeye has a pea-comb, small wattles, heavy bone structure, short beak and a dewlap between his wattles (all indicating an Aseel (Asil) or Cornish blood as the Asian Game)-- today's Buckeyes have an infusion of Dark Cornish in the not-so-distant past. It is not known for sure whether the originator used Cornish in the Buckeye's development (but again, no doubt that today's Buckeye has Cornish blood). The Buckeye was developed as a winter egg layer as well.
The RIR has a finer bone structure, no dewlap. a Single or Rosecomb, larger wattles. The Asian Game influence is not the heavier game like in the Buckeye. The RIR is less of a winter egg layer but is a better egg layer than the Buckeye generally (recall the Leghorn used in the RIR development)----The RIR is more rectangular in shape; the Buckeye is more square in shape. Note that Cochins were used in the development of most of the American breeds. IMHO, the Buckeye is a meatier bird.