How does altering the voltage difference between these two structures affect tone?
It doesn't, at least not directly.
How does it affect power?
It does however, affect this.
Do you have a solid grasp of how triodes work? Do you understand how you can change the grid voltage to alter plate current, but that you can also change plate voltage to alter plate current? The amplification factor of a tube is a measure of how much more effective the grid voltage change is than plate voltage change at changing plate current. The grid is more effective because it is physically closer to the cathode than the plate.
Take a triode, and slap an additional grid between the existing grid and the plate. You now have a screen grid (G2). It is closer to the cathode than the plate, but no where near as close as the control grid (G1).
Have you looked at the plate characteristic curves of triodes and pentodes (or beam power tubes, like the 6V6)? Triode curves generally slope up and to the right, while pentode and beam tubes have curves which are mostly horizontal. Think about what this really means. The horizontal axis of the graph is plate voltage and the vertical axis is plate current.
With a triode, as we increase the plate voltage (move to the right) with a given grid voltage (one of the curves), the plate current rises. Some grid lines rise more quickly that others, but what this really means is
we can envision the tube as having an internal resistance which is relatively small, so that more voltage pushes more current through the internal plate resistance.
With a pentode, the gridlines (above some amount of plate voltage) are nearly horizontal. This is more strictly true with most beam power tubes (like 6L6, 6V6, KT66, etc) rather than true pentodes (EL34, EL84, etc). This means that the plate current is largely unchanged by even a large increase in plate voltage. If you took a 20M resistor and applied 20v, you get 0.001mA of current to flow through the resistor; apply 200v and you get 0.01mA, or just 0.09mA more. Pentodes don't have an infinite internal plate resistance, but it is very high (like several megohms). Ultimately, this means that above some critical value, the
plate voltage has very little impact on plate current.
Note that most data sheets show pentode plate characteristics at a single set screen voltage.
Some datasheets have triode curves for pentodes and beam power tubes. The curves look an awful lot like typical triode curves. Triode operation is when the screen is connected to the plate. Now, plate current rises in a typical manner when plate/screen voltage rises. What does this mean? The the voltage on the screen of these tubes impacts the plate current the way that plate voltage impacts plate current in a triode.
Most data sheets show plate curves for only a single screen voltage. Here's a
rare exception for the 6L6GC. Pick any plate voltage for the 0v gridline on the top graph on page 4, and compare it to the same plate voltage on the 0v gridline for the top graph of page 7. These two graphs show characteristics for a screen voltage of 250v and 400v, respectively. The higher screen voltage always results in a higher plate current for a given grid voltage.
You can also see this effect, for the 0v gridline only, shown in the top graph of page 6. That graph allows a designer to quickly see the peak plate current (occurring when the grid momentarily reaches 0v) at various screen voltages.