What was the difference between the Black and Tan republicans and the lily white Republicans

The most fundamental difference Black-and-tan faction of the republican party and lily-white faction was the color of skin. Black people belonged to the former, while white people to the latter.

Presence of these factions in the Republican party of the South corresponded to the historical period between the 1870s to the 1960s. Black-and-tan was naturally numerous in the states with large black population.