When/why is the solution for an RLC circuit differential equation actually useful?
There are different reasons you'd want to know the time or frequency domain response.
If you are making a relaxation oscillator, or any circuit where you want to time or delay something based on the time something charges/discharges, you'd want the time domain solution. If you are looking at how fast a circuit responds to a sudden change in a digital input, you'd want the time domain solution.
If you wanted to know how a filter responds to different frequencies, you'd want the frequency domain solution.