Are G sharp (G#) and A flat (Ab) The Same Notes in Piano?

Yes G# & Ab notes are the same. They are enharmonic equivalents to each other. It is clear that it is that black key on the piano, which you see between G and A note. It is the note by sharpening G note (therefore G#) or flattening A note (therefore Ab). It is very important to understand when to call it with G# & when with Ab. As, you can not use them interchangeably.
Remember that in a diatonic scale, the alphabets used for naming notes should not repeat, and thereby no alphabet can skip. This is very obvious to understand in C major scale, where all white keys with no flats and sharps. C Major scale

C Major Scale: C D E F G A B C
The root note is not repeated until you complete the octave at the end. This is as per the rule of naming scale notations.

However, in case of C# Major scale we must use the following notes with the alphabets as follows:
C# Major Scale: C# D# E# F# G# A# B# C#

Notice that the scale notes must not have any repetition of the Alphabet. The rule is that each scale has each letter once. Thus, denoting Ab instead of G# will cause no sharpening or flattening of G and the repetition of the A one flattened and other sharpened.  Therefore it has G#

Bb natural minor scale

Bb C Db Eb F Gb Ab Bb

Here, denoting G# instead of Ab will have no use of letter ‘A’. We must use each letter once.

Here is the list of the scales with the correct use of each letter once rule.

C Major: C D E F G A B C

C# Major: C# D# E# F# G# A# B# C#

Db Major: Db Eb F Gb Ab Bb C Db

D Major: D E F# G A B C# D

D# Major: D# E# F## G# A# B C# D#

Eb Major: Eb F G Ab Bb C D Eb

E Major: E F# G# A B C# D# E

F Major: F G A Bb C D E F

F# Major: F# G# A# B C# D# E# F#

Gb Major: Gb Ab Bb Cb Db Eb F Gb

G Major: G A B C D E F# G

G# Major: G# A# B# C# D# E# F## G#

Ab Major: Ab Bb C Db Eb F G Ab

A Major: A B C# D E F# G# A

A# Major: A# B# C## D# E# F## G## A#

Bb Major: Bb C D Eb F G A Bb

B Major: B C# D# E F# G# A# B

The notes with bold and underline mean that there will be a lot to write on the staff notation for denoting the corresponding scales due to the need for using 2 #s. Therefore, usually, it is avoided avoided take those scales. For example, A# major scale is avoided, its enharmonic equivalent scale Bb major is taken.