You use a Schmidt-trigger (S-T) when de input signal is not going directly from 0 to 1 (or 1 to 0 ) but slowly or with a 'bump'. The S-T prefents false data transitions.
See picture:
(input signal)
(output signal without S-T)
(output signal with S-T)
Sometimes, the input voltage is not ground or supply level. For inverter, it will be a large quiescient current while it is not the case for schmitt trigger.