I'm far from an expert and have only really done fixed bias myself... so again, pinch of salt.
What I understand is that fixed bias gives you more control.
1) You can vary the bias by twiddling a pot rather than swapping out resistors (although I suppose there is nothing stopping you from using a variable resistor in cathode bias).
2) As the bias is set externally it does not vary with loading like a cathode bias does so it keeps things more consistent (good or bad depending on what you're looking for).
"Cathode bias often lends a natural compression or 'squishiness' to the sound, due to the increase in bias voltage when one valve enters Class B conditions, though the larger the bypass capacitor, the less will be this effect. A small capacitor (less than 100uF say) also increases non-linear distortion, which may be significant in hifi. Using a very large capacitor (greater than 470uF say), or using no capacitor at all, reduces this effect.
Fixed bias on the other hand, remains the same at all times. This allows maximum output power to be developed, and the reduced compression gives faithful transient response, or a stiff or 'barking' overdriven sound.
Furthermore, there is no reason why we cannot use a little fixed bias and cathode bias simultaneously to achieve the desired mix of compression and 'bark'. "
If you're interested, this is the method I used to generate my bias voltage... and to be honest, I just set it to the nominal value from the schematic and left it there. It sounds pretty sweet so I haven't been inclined to fiddle.
Out of interest, one nice side effect of using cathode bias is that you can use VVR to do power scaling and get some really good tones at low volume.