Transposing is the term for changing the key of a song as your describe.
Theoretically the music will be the same. The same in terms of the relative relationships of rhythm and the intervals that make chords and the melodic line.
But as a matter of orchestration, of timbre, range does matter. It matters a lot. As chords are played in lower and lower ranges they become "muddier." Acoustic guitar in the open string range is fairly deep and resonant, but put a capo on even the fifth fret to raise the pitch a fourth and it can start to sound thinner and bright, sort of like a mandolin or autoharp. Each instrument has certain "colors" associated with different ranges and those are often important choice in a song arrangement/recording.
Imagine a pop song by a singer with a high range, like Mariah Carey, sung in a lower octave by a bass. It probably would not have the same effect as sung in a high range. It could even sound like a comic parody!
Transposing down a fifth - what you described going from
Gm - is roughly the difference between the four basic voice types, so it's like dropping a tenor to a bass. That probably isn't too big a change for the vocal part. Chord voicings in the accompaniment could be adjusted to keep things in roughly the same range for guitar or keyboard. (Example, a bass of
G3 up to
D4 could be transposed down a fifth by playing
C4 down to
G3, the letters transpose a fifth down, but the octaves are juggled to stay within the same general range to maintain timbral color.)
You might consider if dropping a third would be enough to make it comfortable to sing. That would be a change sort of in between vocal ranges, like tenor to baritone which is between tenor and bass.