The parametrization $t\to (t, t^2, t^3, t^4)$ is a closed immersion of the affine line $\mathbb A^1$ to $\mathbb A^4$ (it is the restriction of the $4$-uple embedding of $\mathbb P^1$ to $\mathbb P^4$). So we have a morphism $Y\to Z$ which is clearly an isomorphism outside from $(0,0,0,0)$ because $t$ is invertible there.
At $0$, the second parametrization $f : \mathbb A^1\to \mathbb A^4$ is not a closed immersion. The corresponding ring homomorphism is
$k[x,y, z, w] \to k[t], \quad x\mapsto t^4, y\mapsto t^5, z\mapsto t^6, w\mapsto t^7.$ When localization at $(0,0,0,0)$ the image of the maximal ideal $(x,y,z,w)$ doesn't generate the maximal ideal $(t)$ of $k[t]_{tk[t]}$.
The morphism $Y\to Z$ is birational but not isomorphic. It is in fact the normalization of $Z$.