how does running a 4ohm cab on the 8ohm tap affect the amp (200s) and sound quality (tonal qualities and volume)?
It should generally perform at less than spec across the board.
Most tubeophiles agree that a mismatch of <100% is acceptable from a damage control perspective; i.e. the transformer should be able to handle it, and while it won't do anything
good for your output tubes they should also be able to hang, albeit with an abbreviated life expectancy.
It's not an exacting science. The impedance of a cabinet isn't a fixed quantity; the actual impedance fluctuates based on frequency, wattage, air resistance, etc. A cab rated at 8 ohms may actually show the output transformer <6 to >10 ohms for brief periods in actual use. A speaker rating of 8 ohms is sort of a mean average, and the output transformer buffers these load fluctuations in order to match the amplifiers output stage impedance as closely as possible.
Work the transformer too hard and it
will fail, and almost definitely take some of it’s sibling components with it.
So... while it should "work", your 200S mismatch scenario would be testing the integrity of the amplifiers design parameters, and you should expect performance less than design specifications accordingly.
But why the dilemma? Your 200S has a 4 ohm tap you can use.