Should I give up piano and violin because of child prodigies? As I watch them, it makes me feel like somewhat insignificant.

Quit? Because others are better than you? Miss out on all the enjoyment of music simply because others are better than you? Seriously? Who cares? There are no trophies for the fastest, etc. in music. Thankfully. The pure novice through the seasoned music vet can all enjoy music by themselves, with others etc. Tons of musical genres, buckets of great song, or make up your own. Music is simply where it’s at. 

Quit? lol! Just not an option. I admire those better than me. I enjoy their music. Then, go play my own, inspired by them.