There is a clear and more specific answer (with reference moreover!) here, despite the different question: http://math.stackexchange.com/a/18496/84625
In short $\operatorname{Aut}\left(\left(\mathbb{Z}_p \times \mathbb{Z}_p\right) \rtimes \mathbb{Z}_p) \right) \cong \operatorname{AGL}(2,p)$ while $\operatorname{Aut}\left(\mathbb{Z}_{p^2} \rtimes \mathbb{Z}_p) \right) \cong \mathbb{Z}_{p}^{*} \rtimes \mathsf{AGL}(1,p)$
where $\operatorname{AGL}(n,p)$ is the General Affine Group and $\mathbb{Z}_p^{*}$ is the dual of $\mathbb{Z}_p$.