The SR-71 was made with not tightly fitting fuel tanks and black paint, etc. to deal with aerodynamic heating. The XB-70 flew at essentially the same altitude and speed range, and was designed with not so leaky tanks and pure white painting. Illustrations with temperature data showing nearly the same temperature on the skin of the planes. So how it is possible that these two planes designed to fly in similar circumstances are so differently handling with the heating?
While black surfaces emit heat more efficiently than lighter colours, this was not the only reason for the colour of the Blackbirds.
Earlier iterations of the A-12, the predecessor of the SR-71, were mostly bare titanium externally. The black coating added later was not just any high temperature paint, it was a special compound containing small metallic particles to reduce the reflectivity of the surface in the radar spectrum.
The Blackbirds handled the heat expansion by corrugation of the flat surfaces. This added necessary rigidity to avoid buckling of the panels that would have ruined aerodynamics.
The leaky tanks of the SR-71 were a consequence of weight saving measures more than anything else. To save weight, the fuel tanks were integral to the structure. Bladders or sealants could not be used due to the heat, so the slight gaps left to allow for thermal expansion leaked when the airframe was in ambient temperature.
The XB-70 construction was different from the Blackbirds'. It was made of stainless steel honeycomb sandwich panels, which are inherently rigid, and did not need corrugation or expansion gaps. As for the color, had the XB-70 ever matured into B-70, it would have most likely been painted with the same black compound as the Blackbirds were. The white coloured Valkyries we have seen are white, because they are/were prototypes. The program was cancelled before the aircraft was ready for service.
The color of the aircraft was decided primarily based on the military mission, not for temperature. Black helped the SR-71 blend in better at night and high altitudes, and the paint was also designed to make it less visible to radar. White was chosen to protect the XB-70 from the bright flashes of its nuclear weapons.
While the color may have made a difference for temperature it probably wasn't critical. Darker colors radiate heat better, but they also absorb more heat radiation. The primary means of addressing the temperature issue was using titanium for leading edges and circulating fuel to cool hot surfaces.
Also, the critical limit for the SR-71's speed was not necessarily the structural temperature, but the engine compressor inlet temperature.
As for the fuel tanks, the XB-70 wasn't free of leaks either.
Also note that the aircraft were not equally proven designs. There were only 2 XB-70 prototypes that flew for just over 3 years, while there were 32 SR-71s built (plus 18 of the related variants) that continued to fly for 45 years. The XB-70's fastest flight reached Mach 3.08 while the SR-71 reportedly reached as high as Mach 3.5.
So how it is possible that these two planes... are so differently handling with the heating?
Different materials built in different ways.
The only materials able to sustain high temperature without losing their structural strength are titanium alloys, stainless steel and nickel alloys.
(Specific strength versus temperature, source)
The Blackbird mainly used titanium for its structure (and some special kind of plastic to reduce its radar signature), as explained for example in this answer.
The X-15 used nickel alloys to resist flights at Mach 7.
North American Aviation engineers used a different approach to tackle high temperature and expansion when designing the XB-70. They used titanium alloys for the substructures (some 9% of its structural weight) and the much cheaper and easy to work stainless steel for the rest of the structure (some 70% in weight). Anyway the stainless steel was not used in the form of simple panels like in a standard aircraft (Blackbird included) but they arranged it in "brazed stainless steel honeycomb sandwich panels": basically two sheets of very strong stainless steel alloy with a honeycomb structure between them, all brazed together.
(Cropped image from the original on USAF museum)
These panels were very light (since they were made up of thin sheets), strong (since they had a big total thickness) and insulating (since there was a lot of air between the two sheets). But getting them right was not easy: skin detachments at high speed was initially a big problem with pieces as big as 1 meter detaching and hitting other structures. But eventually this problem was (almost) completely solved improving the fabrication process.
Fuel tanks were sealed with a special rubber and pressurised with nitrogen to avoid auto-ignition: gas-tightness also proved to be a challenge at the beginning but was eventually mastered too, with several layers of rubber applied and cured in a specific way.
Note that the two XB-70 built logged only a total of 1:48 hours at Mach 3+: basically nothing in respect to the Blackbird, so it is not possible to say if this construction technology would have been better or worse than the Blackbird's one.
But from a metallurgy point of view, the Valkyrie taught a lot of valuable lessons still in use today.
Regarding the colour: I agree with the answer of @fooot, i.e. it is normally chosen according to the mission. The Blackbird was mainly flown at night and a dark colour (which was actually blue and not black) made sense with the bonus point of helping in radiating heat. The Valkyrie was supposed to drop nuclear bombs so it was painted in a bright "anti-flash" colour.