For a series of examples in any dimension, take a degree $2k$ hypersurface $H_{2k} \subset \mathbb{P}^n$. Correspondingly, there is a double cover $X_{2k} \to \mathbb{P}^n$, branched over $k$, so $X_{2k}/\mathbb{Z}_2 \cong \mathbb{P}^n$.

If $k \neq h$ then $X_{2k}$ and $X_{2h}$ are not even homeomorphic, for instance because they have different topological Euler number.

For $n=1$ we recover the example of hyperelliptic curves given in Nick L's answer.

For a series of examples in dimension $n=2$, take a degree $2k$ hypersurface $H_{2k} \subset \mathbb{P}^2$. Correspondingly, there is a double cover $X_{2k} \to \mathbb{P}^2$, branched over $k$, so $X_{2k}/\mathbb{Z}_2 \cong \mathbb{P}^2$.

If $k \neq h$ then $X_{2k}$ and $X_{2h}$ are not even homeomorphic, for instance because they have different topological Euler number.

For a series of examples in any dimension, take a degree $2k$ hypersurface $H_{2k} \subset \mathbb{P^n}$. Correspondingly, there is a double cover $X_{2k} \to \mathbb{P}^n$, branched over $k$, so $X_{2k}/\mathbb{Z}_2 \cong \mathbb{P}^n$.

If $k \neq h$ then $X_{2k}$ and $X_{2h}$ are not even homeomorphic, for instance because they have different topological Euler number. For $n=1$ we recover the example of hyperelliptic curves given in Nick L's answer.

For a series of examples in any dimension, take a degree $2k$ hypersurface $H_{2k} \subset \mathbb{P}^n$. Correspondingly, there is a double cover $X_{2k} \to \mathbb{P}^n$, branched over $k$, so $X_{2k}/\mathbb{Z}_2 \cong \mathbb{P}^n$.

If $k \neq h$ then $X_{2k}$ and $X_{2h}$ are not even homeomorphic, for instance because they have different topological Euler number. 

For $n=1$ we recover the example of hyperelliptic curves given in Nick L's answer.