The Concorde was different from regular jet airliners because it was a supersonic aircraft, and that single fact explains almost everything about why it flew so much higher. The problem with subsonic aircraft is that as the altitude gets higher, their stall speed gets higher and the speed of sound gets lower. The stall speed is the minimum speed the plane can fly at without stalling and falling out of the sky in an uncontrolled fashion.
Above a certain altitude, the minimum stall speed exceeds the speed of sound, and to go higher than that, the aircraft must exceed the speed of sound. The altitude at which subsonic aircraft cannot fly without stalling is often called Coffin Corner, a place where pilots do not want to go. To go higher than coffin corner, they need to go supersonic, and regular airliners cannot do that. Concorde could.
The Concorde never really cruised at 60,000 feet in the traditional sense. After takeoff, once they reached the point to initiate Mach 2 flight, they would set the throttle to max and kick in the afterburner. Once Mach 2 was achieved, the afterburners were turned off, however the throttle remained at max throughout the rest of the cruise phase of flight. The Concorde would then settle into a very slow cruise climb, and as it burned off fuel it would get lighter and climb ever so slowly until it reached the top of climb at around 60,000 feet. Because of this unusual cruise and climb pattern, the Concorde had its very own air corridors over the Atlantic ocean reserved just for them.
When you consider that drag is proportional to the square of speed, it becomes very apparent that in order to minimize the fuel consumed during cruise, you need to minimize the drag at the desired speed. This is achieved by flying at an altitude where the air density is low but still suitable to provide the desired lift at the desired speed. At high altitudes the air is thinner, allowing the Concorde to fly faster and reduce fuel burn. At low altitudes there was concern about skin heating and environmental concerns, and when flying at lower altitudes the Concorde would be subjected to excessive drag and friction created by the heavier air.
Regular airliners top out at about 41,000 to 45,000 feet, mostly due to pressurization limits of the aircraft structure. At above 41,000 feet, time of useful consciousness after a sudden decompression is so low that occupants will not even be able to don their oxygen masks. That was the critical safety reason for keeping subsonic airliners at lower altitudes. The air also produces heat in the form of skin friction, and it is not good for the skin or leading edges to overheat, which is another reason the Concorde needed to be as high as possible where the air was thin enough to reduce friction heating.
Available power, fuel consumption and skin temperature are the major factors that set practical altitude limits for any aircraft. The higher you fly, the faster you can potentially go and the less fuel you will burn. Sixty thousand feet is close to the practical ceiling for air-breathing vehicles, and while the SR-71 could reach 80,000 feet, it needed to fly at Mach 3 to do it, which is not practical for a passenger aircraft. For the Concorde, despite all the various trade-offs, 60,000 feet was its best compromise ceiling.