What is the difference in radiated power between a lossless antenna with gain and an isotropic radiator driven by the same power?
Why Antenna gain is not amplification; a passive antenna cannot create power. A lossless antenna radiates 100% of the power delivered to it, exactly as a lossless isotropic radiator does. What gain describes is how that same total power is redistributed, concentrated into some directions at the expense of others, so the field is stronger in the main lobe and weaker elsewhere while the integral over the whole sphere stays constant.
Watch out The idea that the directional antenna radiates more power confuses power density in one direction (which does rise by the gain factor) with total radiated power, which cannot exceed what the transmitter supplies.
Gain redistributes, it does not create. Same watts in, same watts out; only the shape of the pattern changes.