Two representations of $R_z$ are equivalent if they are the same modulo only a global phase.
$$
\begin{pmatrix}1 & 0\\
0 & e^{i\theta}
\end{pmatrix}
=
e^{+i\theta/2}\begin{pmatrix}e^{-i\theta/2} & 0\\
0 & e^{i\theta/2}
\end{pmatrix}
$$
If you apply this gate to any state $|\psi\rangle$, the only difference in the outcome is a global (constant) phase of $e^{i\theta/2}$. This cannot be detected by any measurement.
For example measurements on the state:
$$
|\psi\rangle = \frac{1}{\sqrt{2}}\left(|0\rangle + |1\rangle \right)
$$
will result in $|0\rangle$ with a probability of $\left|\frac{1}{\sqrt{2}}\right|^2 = \frac{1}{2}$, and
will result in $|1\rangle$ with a probability of $\left|\frac{1}{\sqrt{2}}\right|^2 = \frac{1}{2}$.
Now consider the state:
$$
|\psi\rangle = \frac{e^{i\theta/2}}{\sqrt{2}}\left(|0\rangle + |1\rangle \right)
$$
Measurements will:
result in $|0\rangle$ with a probability of $\left|\frac{e^{i\theta/2}}{\sqrt{2}}\right|^2 = \frac{1}{2}$, and will
result in $|1\rangle$ with a probability of $\left|\frac{e^{i\theta/2}}{\sqrt{2}}\right|^2 = \frac{1}{2}$.
Two gates that are equivalent up to a global phase (one that's the same for all components) are equivalent for the purpose of anything you will be able to detect by measurement.