What filter is better with -1/3 2/3 -1/3 or with -1/4 1/2 -1/4 coefficients?
Both filters can be re-written in the form:
\begin{equation}
H(z) = \alpha ( 1 -2z^{-1} +z^{-2} )
\end{equation}
Clearly, in the first case, $\alpha=-\frac{1}{3}$ and in the second $\alpha=-\frac{1}{4}$.

In other words, the shape of the filter's frequency response is the same, the only difference is the scaling.

I can think of two reasons of why the second filter may be preferred.


In the case of the second filter, the maximum gain of the filter (occuring at the Nyquist frequency) is normalized to $1$, whereas the first filter has a maximum gain of $\frac{4}{3}$.
The other more compelling reason however, is probably that the coefficients of the second filter can be expressed as powers of two ($\frac{1}{4}=2^{-2}$, $\frac{1}{2}=2^{-1}$) and therefore multiplication by these coefficients can be quickly implemented as bitwise operations on a computer or digital signal processor.