The bridge is the key point of connection between the strings and the body, but the tailpiece is an essential part of this. The main effect of upgrading this will be to change sustain and tone.
You may also notice an improvement in tuning stability - if you are upgrading from a poor quality bridge/tailpiece.
The actual effect varies a lot between an acoustic, semi-acoustic and solid bodied guitar, and also between through-neck and bolt-on guitars, however unless there is a specific sound you are after, I wouldn't see this upgrade as that useful - it is very minor compared to most of the other modifications you could make.
My suggestion: don't do this unless you have a broken or poor quality tailpiece, or want to change the finish. You'd be better off saving for another guitar if you want a different sound.