Why use Teichmuller representatives? In p-adic mathematics, what is the advantage of using Teichmuller representatives over using just the numbers 0,1,2,...,p-1 ?
In either case, the norm is the same.
In either case, all the points are in the center.
Is is fair to say: "Teichmuller representives, just like the digits 0,1,2,...,p-1 , are just names of elements that are different in name but in no other characteristic.  Although p-adic arithmetic seems more complex using Teichmuller representatives, more advanced mathematics is easier and more robust."
 A: The Teichmüller lift can be seen as a map of multiplicative monoids from $\mathbb{F}_p$ to $\mathbb{Z}_p$, that is the unique multiplicative section of the mod $p$ reduction map.  In particular, the lift produces roots of unity from nonzero inputs.  You cannot say that about the integers from $1$ to $p-1$ (which tend to have infinite order). 
Unlike writing integers up to $p-1$, taking the Teichmüller lift is a method that readily extends to more general settings.  As KConrad mentioned, there is a more natural way to view the Teichmüller lift, as the embedding into the zeroth spot in the ring of Witt vectors.  This works not only for the field with $p$ elements, but for any characteristic $p$ ring.  In particular, finite extensions of $\mathbb{F}_p$ have Teichmüller lifts to unramified extensions of $\mathbb{Z}_p$.  Natural operations like the Frobenius lift and Verschiebung can be expressed concisely in the Witt vector setting.
I mostly disagree with the quotation you offer.  Integer representatives are better-suited to $p$-adic addition than Teichmüller representatives, and this makes explicit computations with polynomials and power series more tractable.  Multiplication is handled by both types of representatives with roughly equal ease.  If you want to prove a theorem (involving algebraic structures in the $p$-adics), you are likely to be better off with Teichmüller representatives.
