What is the difference in radiated power between a lossless antenna with gain and an isotropic radiator driven by the same power?

The total power radiated by any lossless antenna is exactly the same as the total power fed into it, regardless of its shape or design. Antennas are entirely passive devices and cannot magically create or amplify raw power. When comparing a lossless antenna with high directional gain to an isotropic radiator (which radiates equally in all directions) driven by the same transmitter power, the total amount of energy released into space is mathematically identical. The term "gain" simply means the antenna focuses that existing power in a specific direction at the expense of other directions, much like the reflector in a flashlight focuses a bulb's light into a narrow beam. The total power radiated remains entirely unchanged.