When you are playing fretless string instruments, especially bowed instruments in small groups, you become very sensitive to these differences. I will not quantify them as there are already other answers on this area. When I was young, I was told the comma model of the occidental scale and I think it is a good first approach of these issues in most classical music, even if it is theoretical and limited. It gives you an easy way to remember the relative placements of accidentals.
Musically there are at least three musical dimensions where it is felt, one I would call melodic, another harmonic and still another timbral for lack of more nuanced words.
When you have a melodic lines that use accidental alterations, or hesitate between several modes and tonalities (this is quite common), the exact sensation you produce with your instruments is quite dependent of the exact intonation. There is a rhetoric quality to some music and it is enhanced by careful intonation. Depending on the period/school of the piece you may need to be very careful about the way you play accidentals. In fact certain qualities that are intuitively felt by listening to very good interpretation of music are due to a careful respect of these differences.
When playing double-stops (on a single instrument) or chords across a group, you cannot afford to play a Gb like a F# together with, say, a natural D. It is usually the same finger but not exactly the same place and angle of your finger on the fingerboard. It does not sound the same, but here again, chords are not isolated in a music piece. They should be heard in succession and only their contrast is really meaningful. As starting player usually make larger intonation errors than two or three commas, this is something which is usually treated after several years of study but could be intelligently made before.
The sound quality of your instrument can be different. A small change in pitch gives a different balance of resonance of the body of the instruments and of the other strings. Another direct aspect is that you cannot play for instance a Ebb on the D string of a cello. You have to find it by playing on the lower G string to be in tune. So it sometimes change fingerings. It is true with wind instruments such as the transverse flute where you have alternate fingerings to sound "more flat" or "more sharp" for certain notes in addition to the intonation changes that good player can create with their breathing technique.