Confused about Sharps and Flats
General mountain dulcimer or music discussions
It depends on what temperament is being used, as Skip also mentioned. In equal temperament, G# is the same as Ab. There are other temperaments (systems of tuning used at different points of history and/or in different cultures) where G# may not be the same as Ab.
In addition to that, players of fretless or other instruments capable of microtones may choose to inflect notes depending on what follows them. For example, leading tones (like G# in the key of A) might be pushed sharper in order to accentuate that leading to the tonic.
To complicate things further, due to acoustical physics of various instruments, there are often tuning peculiarities regardless of what temperament the instrument maker was going for. Clarinets, for example, overblow an interval of a 12th- but some fingerings end up producing wider or narrower 12ths, even on high end instruments. It's up to the player to be able to compensate for that. And of course some ears are more sensitive than others to even notice.
More than you probably wanted to know for your question!