There are multiple factors that affect an aircraft based on its cruise altitude.
The cruise altitude directly affects the aircraft pressurization and aerodynamics. In order to keep the cabin altitude around 6000 to 8000 feet, the fuselage would have to withstand a higher pressure differential. This would require more material, and make the plane heavier. Newer materials used on the 787 and A350 handle these loads better, which allows a higher pressure differential and larger windows.
The altitude also affects the aerodynamics. Aircraft typically cruise at a certain Mach number at high altitudes. The indicated airspeed for a certain Mach number gets lower with higher altitude. This can result in lower drag, but poses challenges as the airspeed drops lower. See: What determines the maximum altitude a plane can reach?
Also, see the answer by Peter Kämpf for more detail about the aerodynamic aspects.
At higher altitudes, less air is available for the engines, reducing the available power. Along with this there are efficiency benefits. See: Why do jet engines get better fuel efficiency at high altitudes?
As the answer by Ghillie Dhu explains, those benefits stop increasing around FL360.
The lack of breathable oxygen and increased pressure forces on the fuselage caused by the lower air pressure also lead to greater risks, resulting in tighter regulations on aircraft intending to fly higher. These regulations will add additional cost to flying higher, affecting decisions on service ceilings.
(d) The airplane structure must be designed to be able to withstand the pressure differential loads corresponding to the maximum relief valve setting multiplied by a factor of 1.33 for airplanes to be approved for operation to 45,000 feet or by a factor of 1.67 for airplanes to be approved for operation above 45,000 feet, omitting other loads.
So for planes operating over 45,000 feet, the structure must be able to withstand an even higher pressure load factor.
(d) The oxygen flow rate and the oxygen equipment for airplanes for which certification for operation above 40,000 feet is requested must be approved.
Planes operating over 40,000 feet must get special approval for the oxygen system.
(2) The airplane must be designed so that occupants will not be exposed to a cabin pressure altitude that exceeds the following after decompression from any failure condition not shown to be extremely improbable:
(i) Twenty-five thousand (25,000) feet for more than 2 minutes; or
(ii) Forty thousand (40,000) feet for any duration.
The cabin altitude is not allowed to exceed 40,000 feet under any condition that isn't extremely improbable. The higher the plane flies, the more difficult this will be to certify.
Business jets tend to have higher cruise altitudes, because the smaller structure is easier to reinforce for higher pressures, and typically sees fewer load cycles. Efficiency is also not as important as speed and comfort, so some weight can be spared for the stronger fuselage. The higher altitude also affords more flexibility in routing since most other traffic including commercial airliners are cruising at lower altitudes. Business jets may also be certified under less-stringent regulations.