There is a discussion on it here that's worth reading but in short the requirements were just different.
A few of the key points, The black color on the SR-71 offered some night camouflage in addition to its heat dissipation
The Concorde had an Aluminium airframe while the SR-71 had a primarily titanium airframe which could lead to different coating types.
The Concorde was a commercial aircraft many of which are often painted white thermal benefits aside they may have simply been keeping with what they usually did.
The SR-71 flew substantially faster than the Concorde and had different thermal requirements.
The hottest point on the Concorde was the nose 127°C which was actually cooled by fuel being pumped through as a coolant.
The hottest point on the SR-71 was the cockpit window which cooked in at 327°C again, very different requirements thermally.
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Concorde's average skin temperature was 92°C (365K).
Calculating the black body radiation using the Stefan Boltzmann law we get 1006W/m².
This the maximum heat flux possible with perfect radiation, and very similar to the heat flux of solar radiation, which is also about 1kW/m² at the earth's surface in the absence of clouds (and a bit higher at Concorde's typical cruising altitude.)
However, as others have pointed out, at these temperatures Concorde would radiate in the far infra red, and it is perfectly possible to have a selective paint that appears white in the visible region (reflecting much of the heat of sunlight) while also radiating in the infra red region.
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The Blackbird was black so it could absorb radiation, not so it could emit radiation.
Look at the Blackbird's predecessor, the A-12, it is fairly easy to find pictures of the A-12 with polished or partially polished finish.
The Blackbird and A-12 moved to a radar absorbent black finish for reasons of observability, flying at the edge of space means a darker surrounding than lower in the atmosphere so the camouflage requirement is different than say a fighter jet.
Any thermal emissions that do occur do so well outside the visible spectrum.
To reach the point at which it is beneficial to thermally dissipation for the aircraft to be painted black is to reach the point that the aircraft emits more energy as radiation than it absorbs in the visible spectrum.
At that point the black aircraft would be brighter than a white aircraft.
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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.
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Concorde was a civilian commercial vehicle.
It's mission is not 100% flying, but also embarking and disembarking of passengers, waiting for ATC permission, etc.
This means, it spend considerable time, on the ground, on low power, not heated by the airflow - but heated by the sun.
For this part of the mission, white paint is vastly superior.
SR-71 was a military aircraft and it was optimized for one thing only - the flight.
Sunbathing at the airport was not an issue, it was kept in a hangar for as long as possible anyway.
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It's well-known that the stated reason the Concorde was painted predominantly white  was to mitigate heating problems .  However, given that the source of the Concorde's thermal woes wasn't excessive exposure to solar radiation, but, rather, direct conduction and convection of compression heat, I'm confused; in that case, shouldn't the Concorde have been painted black (or nearly black),  like the SR-71 , to better radiate heat away?
Why was the Concorde painted white and not black?
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<concorde><livery><paint>
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S2: The Concorde had an Aluminium airframe while the SR-71 had a primarily titanium airframe which could lead to different coating types.  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. Moreover, the Concorde was a commercial aircraft many of which are  often painted white  thermal benefits aside they may have simply been keeping with what they usually did.    The main reason is the heat absorption and heat emission of each material. Concorde's aluminium airframe is favoured by a lighter color scheme. It is more advantageous to paint an aircraft black only when it emits more energy as radiation than it is able to absorb, within the visible spectrum. Another reason is perhaps that the Concorde was a commercial airplane, which are usually painted white.
The main reason is the heat absorption and heat emission of each material. Concorde's aluminium airframe is favoured by a lighter color scheme. It is more advantageous to paint an aircraft black only when it emits more energy as radiation than it is able to absorb, within the visible spectrum. The Concorde was a commercial airplane, which are usually painted white.