There can be the perception of difference in power depending on how overload is managed by each particular design. That said, some designs are intended to be driven into clipping and clip more gracefully than others. This is (or can be) completely independent of class.
Efficiency is defined by Power out/Power in x 100%. Therefore, using my previous example of a 90% efficient typical class D amp and a 60% efficient class G/H amp, if the amps were both rated at say 500 watts rms, at rated power the class D amp would be drawing 500/90% = 500x.9 = 555 watts. The class G/H amp would be drawing 500/60% = 500/.6 = 833 watts. Therefore, it would be correct to conclude that this class D amp is more efficient than this class G/H amp by (833 watts - 555 watts) = 278 watts (the difference is waste heat dissipated by the amp).
This is factually incorrect. A tube amp rated at the same power and same distortion level as a solid state amp will have exactly the same output.
What happens is that IN GENERAL, most tube amps will clip more gracefully and in a more pleasing manner than most solid state amps. This means that it's possible to drive the amp harder into clipping with a tube amp and still find the sound acceptable. This was especially true of older school solid state amps that clipped particularly ungracefully, but in recent years, designers of higher quality solid state products have closed this gap by developing ways to mimic the more graceful clipping of tube amps using a variety of soft clip, compression, limiting and dynamics management methods.
This is all part of the perception which can, when misunderstood, lead to incorrect conclusions even when there are real, perceived differences.