The piano, together with the organ and harp are unique musical instruments in the sense that each hand can play notes independently of the other hand. If you think about stringed instruments such as guitar and violin, the left hand usually determines what note or notes are played while the right hand determines the quality of sound such as if it is bowed (in the case of the violin) or strummed (in the case of the guitar) or plucked. It is also the right hand that determines how loud or soft the note is played. When it comes to instruments that are blown into such as trumpet, trombone, flute saxophone and clarinet, only one note can be produced at a time with both hands being equally responsible for producing the note (as well as the mouth!).
I can’t comment on which is more important – the right hand or the left hand when it comes to the organ or the harp because I don’t play those musical instruments. But when it comes to the piano I can tell you that the answer is – it depends.
The reason why it depends is because it is the melody or a little musical fragment that is always the most important part and always needs to be heard clearly above everything else, so whichever hand is playing the melody is the most important one. Most of the time it is played with the right hand but sometimes it is in the left hand and sometimes the melody can swap between the right hand and the left hand.
If you are playing music by composer J.S. Bach, you will often find the music is polyphonic which means that there are two or more melodies being played simultaneously or it’s the one melody played against itself, but the starting point of that melody is happening at different times. An example of this is a musical round (and often these are simple folk type songs).
In the case of polyphonic music, it is always important that all parts are heard equally otherwise you could have one part dominating or even drowning out another part.
When it comes to the majority of music, where there is one single melody, whether the right hand or the left hand is playing, it is important that it is clearly heard above everything else. Unfortunately when the melody is in the right hand (which happened in the majority of piano music), there is a natural tendency for the left hand part to drown out the right hand melody part.
It does take concentration and practice but it is always important for a right hand melody to be a bit louder than a left hand part simply because that left handed part is often an accompaniment, a way to enrich and make the sound and give the piece of music a fuller, richer sound.
So when there is a discussion about which hand is more important in piano music, the question should really be about which hand plays the melody because the melody, no matter which hand is playing it, should always be the most prominent part.