The cap in series and inductor in parallel with the tweeter will begin to increase their impedance below the resonance of that circuit (depends on the value of the cap inductor and impedance of the driver at the given freq). The increased impedance attenuates the output of the tweet by cutting the power going to it. The inductor in series and cap in parallel with the mid do the same thing above circuit resonance. The crossover loses power around the crossover freq but is fairly transparent, power wise, well above and well below. With the tweeter and mid paralleled by the crossover, the effect is a fairly smooth division of power between the two.
While the amp is capable of running the same amount of power to the tweet as to the mid, and the crossover usually won't stop it from doing so (barring an attenuation circuit of some sort) most music has a lot less up high compared to down lower. Less energy present in the signal means less power going to the driver. Look at a spectral analysis of a typical song. It slopes fairly significantly from the lows down to the highs. Play pink noise though and prepare for your ears to bleed.