The coarse and fine variable offset controls on your HP 1141A differential probe allow you to adjust the DC offset of the measured signal. This can be helpful in situations where you need to centre the waveform within the oscilloscope's display or align it with a specific reference level.
If the waveform is shifted too high or too low on the oscilloscope screen and you need to bring it back to the centre or align it with a particular reference level, you will need to use the coarse variable offset control.
If the waveform is nearly centred but slightly off, or if you need to align it very precisely with a reference level, you would use the fine variable offset control.