When talking about jazz and classical mouthpieces I believe that you first need to understand the major sound difference between the two genres of music.
In classical saxophone music, players typically have a more focused sound with a strong core.
In jazz music, saxophonists typically have a bigger fuller sound with more grit.
These key determining factors will help explain how the mouthpieces differ. Classical saxophone mouthpieces usually have a smaller tip opening with a focused chamber. These chambers usually help focus the air stream giving the player a more focused sound with a strong core.
Jazz mouthpieces are essentially the opposite of this. They typically have an opened chamber to help the air stream be more spread to help assist with a bigger sound.
I personally believe that having a different mouthpiece for classical and jazz is a must.