With dynamic driver headphones, where the impedance changes over the frequency response, the output impedance of the amplifier determines the final frequency response that you hear. It is for this reason that sometimes people run for example the HD800S on tube amps, along with more second order harmonic distortion which produces a more "euphoric" sound. The high output impedance however is not unique to tube amps and can be reproduced by simply adding a resistor to output section of a solid state amp.

Powerful amps give you more headroom. I'm going to give a simplified example. Let's say in a certain song the drums are 10 dB louder than the vocals. Since dB is logarithmic, you can perform addition and subtraction on the numbers and maintain the difference in relative loudness. An under powered amp tops out at 90dB for a given headphone. If you turn the vocals up so that they are 90dB, the drums are still going to be 90dB because that is the limit of the amp. If you've got a more powerful amp that can drive the headphones up to 110dB, it will comfortably reproduce vocals at 90dB and drums at 100dB.

Amplification is a very complicating topic. You may come across SINAD for example, but they don't tell the whole story. But in general if you get an amp from a reputable brand which delivers high power you are good to go