Fast Time Weighting
Fast corresponds to 125 milliseconds up and down. The needle would be moving much faster if you use analogue, showing you the varying noise levels quickly.
The Fast Time Weighting is usually used for measuring all the sound in your environment, which may vary widely over time. Therefore, you would need to report the range of noise fluctuations e.g. 58 to 64 dB (A). Fast is generally the chosen Time Weighting for noise measurements.
Slow Time Weighting
Slow corresponds to 1000 milliseconds up and down. By slowing down the needle, the noise measurements are easier to read as the data is smoothed out.
By using the Slow Time Weighting, you are recording sound levels at 1 second intervals. This smoother level history can give you a better indication of the average noise level in an environment where it is constantly changing.