Let $ξ$ be a Gaussian white noise on $\mathbb R^d$ ($d=1,2,3$). Let $(ξ_\varepsilon)_{\varepsilon>0}$ be continuous Gaussian processes such that $ξ_\varepsilon\toξ$ as $\varepsilon\to0$, defined by convolving $ξ$ against a mollifier. We consider the asymptotics of the parabolic Anderson model (PAM) with noise $ξ_{\varepsilon(t)}$ for large time $t\gg1$, and the Dirichlet eigenvalues of the Anderson Hamiltonian (AH) with potential $ξ_{\varepsilon(t)}$ on large boxes $(-t,t)^d$, where the parameter $\varepsilon(t)$ vanishes as $t\to\infty$. We prove that the asymptotics in question exhibit a phase transition in the rate at which $\varepsilon(t)$ vanishes, which distinguishes between the behavior observed in the AH/PAM with continuous Gaussian noise and white noise. By comparing our main theorems with previous results on the AH/PAM with white noise, our results show that some asymptotics of the latter can be accessed with solely elementary methods, and we obtain quantitative estimates on the difference between the AH/PAM with white noise and its continuous-noise approximations as $t\to\infty$.