Comparing the control performance of a given closed loop system
The error is a system state's distance from a desired state. If you want a dart to hit bullseye on the dartboard, and you miss, the distance to the bullseye is the error. The RMSE (root-mean-square error) is a way to summarize the error over time, expressing the error as one number. This makes it more easy to compare different cases. For a stationary/steady-state reference signal, such as a sinusoid, if you subtract the mean, it converges to the standard deviation of the error. The sensitivity plot for a closed loop system gives the RMSE directly for a given frequency. Think of a reference signal frequency, look it up on the Bode diagram for the sensitivity, and you know the scaling of the error at that frequency.

The root-mean-square change is a way to measure the power spent on controlling your system. Often you have limited resources available to drive a system to a given state. Think of running vs. walking. If you can slow things down, you reduce the power required. If you draw too much power you can exceed the limitations of a physical system. You might saturate a voltage amplifier or snap an axle... There is a classical trade-off between manipulation power and the smallness of the error.