I've heard it said that, whilst on most instruments these notes are played with the same fingerings/technique/etc., there is a subtle difference.
This isn't specific to this particular note combination, but to all enharmonic equivalents.
What might this teacher have been referring to?
See the section on tuning systems on Wikipedia for some background.
In short, most intervals do not sound best on equally-tempered scales (where the distance between any two consecutive half steps is the same) but on ones where the notes vary in distance. For instance, fifths usually sound the most in tune when the frequencies are in a 2:3 ratio.
Because of this, G♭ and F♯ will often sound different depending on which scale they're being used in and which notes they are played with. As far as I know, G♭ is never higher than F♯, always lower (or perhaps the same, like on a piano).
It depends on the tuning system being used. If you're tuning by perfect intervals, i.e. intervals in which the ratios of the frequencies are in whole-number pairs, then Gb isn't exactly the same as F#.
For example, say you're tuning to A440 and using perfect intervals. Then the E above the A is tuned to 440 * 3/2 = 660 Hz. The B above the E is tuned to 660 * 3/2 = 990 Hz. The F♯ above the B is tuned to 990 * 3/2 = 1485 Hz. Meanwhile, the D below the A440 is tuned to 440 * 2/3 = 293.333 Hz, the G below the D is tuned to 293.333 * 2/3 = 195.555 Hz, and so on.
In the end, and adjusting for octaves, you get that Gb = 366.25 Hz while F# = 371.25 Hz. Not exactly the same, but pretty close. Not close enough not to be noticeable, though.
In Equal Temperament, pitches aren't determined by whole-number ratios. Rather, you use the formula:
frequency = 440 * 2^(n/12)
where n is the number of half-steps above or below the A440 reference pitch. This ensures that enharmonic equivalent notes have the same frequencies, but it also means that no interval is "perfect" in the whole-number ratio sense.
There are the tuning differences, as already mentioned.
Then there is the function difference. If you have an entire piece in D major, using the tones in D major, seeing a D♭ instead of a C♯ would be very awkward.
When writing music, the rules (simplified) are:
Bad spelling of enharmonies is something I see more and more. Probably because music publishing has become more and more easy so less and less skilled people do it, and because many think that a program's auto-transpose function actually works (or they know it doesn't, but they don't care).
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.
As an isolated question, it's sometimes hard to understand why it's important that there is a difference but understanding why there is a difference is an important foundation to Western melody and harmony.
So, imagine you're in the key of D major. The diatonic notes are: D E F♯ G A B C♯. What does F♯ mean? It means the third note of the scale. What does G♭ mean? It means you've taken the fourth note of the scale and lowered it it.
In 12-tone equal temperament, they may sound the same; you may play them the same on the piano or the guitar. But if the function of the note at a particular point in the piece is as the third note in the D major scale, you can only write it F♯ and not G♭. F♯ means something completely different.
It's the musical equivalent of "hear" versus "here". Just because they are homophonic doesn't mean they are the same word. Similarly, in western tonal music F♯ doesn't mean the same as G♭.
