So, lets see how my understanding is...
For a resistive load, an increase in plate voltage will decrease current through the tube, reducing the cathode voltage which causes the current to rise and the plate voltage to stabilize. in other words the resistive load has a feedback loop which stabilizes the plate voltage by changing current.
The CCS will hold the current and cathode voltage constant without the feedback loop.
I would expect that the feedback nature of the resistive load would cause peaks to be somewhat smoothed or dulled due to the feedback loop sort of interfering with the output, reacting, counteracting and stabilizing it. I would expect that the CCS would tighten up and improve the impact of these signal peaks. Plus the improved PSRR.