Good question. There are two main reasons for the Concorde's specific coloring: Heat absorption and heat emission. The key to its color scheme is in the materials used in construction of each plane.
The Concorde was made of aluminum, which emits heat far more quickly and effectively than the SR-71's titanium skin. Contrary to popular belief, the plane will become ductile and lose structural integrity at far lower temperatures than the metal's melting point, and preventing the metal from reaching these temperatures is a key reason for the paint colors of each aircraft.
Aluminum becomes ductile and unacceptably weak at higher temperatures, but would rarely reach them at the Concorde's cruise speed, as the metal reflects/releases sufficient heat so that the paint can can do a more proactive job of keeping heat absorption at a minimum. If titanium had similarly good heat dissipation capabilities then I am confident that it too would have been white.
However, titanium at speeds around the Blackbird's cruise range would absorb far too much heat for it to release, so the black paint would actually be superior in terms of heat dissipation to compensate for titanium's natural characteristics. Short answer shorter, the aluminum on the Concorde didn't get hot enough for the aluminum to lose integrity, so white was the better option in terms of preventing the metal from reaching those critical temperatures.
The SR-71 however could not release heat as quickly as aluminum, so the paint was designed to aid it in that regard. Hope this helps, sorry about the formatting I am in a rush so if there was anything conceptual I left out then please comment.