These two competing solutions both supply the basic power to the audio tubes and output transformers, so only indirectly interact with the sound signal. I have not read anything that explains what tube rectification brings to the party.
Voltage drop across the rectifier, especially when the amp is making power, is one very large reason rectifiers can affect the sound. The other reason is noise.
Solid state rectifiers are often blamed for making noise but its really how the power transformer interacts with the rectifiers. Between the two an electrical resonance can occur which is set into oscillation by the rectifiers turning on and off. The resonance might be at 2MHz but can cause the rectifiers and transformer to make noises much lower in frequency, which can leak into the signal chain and cause IMD. This is known as a 'swept resonance'. Its not something you often run into with a tube rectifier.
But it can be solved with proper application of snubber networks. The best approach is a small capacitance in series with a resistor across the input to the rectifier(s), directly across the output of the transformer.
Once the swept resonance is shut down there's no going back to tube rectifiers as the power supply sags less at higher power levels- which makes for less distortion at those power levels.
An obvious advantage of a solid state rectifier is the capacitances in the power supply can be quite a lot higher. There is often a fairly low limit to capacitor values when tubes are used. There are timing constants involved with these values. What you want is that none of the timing constants in the amplifier circuit be lower than those in the power supply- else the amp can much more easily modulate the power supply, resulting in higher distortion.