Aerodynamic heating will damage or destroy the composite wing of the F-22, especially around the leading edge where compression heating is highest. See here for the temperatures which supersonic flight causes.
Therefore, the F-22 has been restricted to Mach 1.8 for short duration and Mach 1.6 for prolonged flight. More speed will not be needed, anyway.
The internal weapons storage of the F-22 makes it bulkier than the F-15 which causes more wave drag. Also, the engine intakes were not designed to operate at speeds above Mach 2, so pressure recovery in them drops at higher speed. These two factors will affect the top speed of the F-22 negatively, so I would not be surprised if the F-15 is faster even without restrictions on the F-22 speed.