On an upright
piano, the action of the
soft pedal is quite different: pushing the pedal down
advances the hammers in the cabinet by about one third of the way towards the strings.
This has the effect of diminishing the kinetic energy of the hammer throw--which is the characteristic action of any piano--thus, making the tone produced softer, or quieter.
For myself, I do not usually use this, because the effect is marginal at the best of times. For students, this can just be an additional distraction from the notes and all the other things that go into a good performance.
The reason for the difference in the tone, the less sharp attack ferrouslepidoptera mentions, has to do with the felt on the hammer tip--actually a rounded end. When used, the felt will grove and become hard, thus giving a brighter tone to the struck string. When the entire action is moved, when the soft pedal is pushed, then a lessor used portion of the felt strikes the string. This gives a softer, darker tone to the note.
This is why in some composers--I think of Claude Debussy's Serenade of the Doll, in the Children's Corner Suite in paticular--specify forte even when the soft pedal is to be used, just to take advantage of this change of tone.