Consider the :
$\textbf{Underdamped Langevin}$
\begin{align} dX_t&=V_tdt \\ \frac{m}{\gamma}dV_t&=-V_tdt-\nabla \phi(X_t)dt+\sqrt{2D} W_t. \end{align}
I believe $m$ is the mass, $\gamma$ constant of friction, and $D$ diffusion\temperature constant.
$\textbf{Overdamped Langevin}$
\begin{align} dX_t&=-\nabla \phi(X_t)dt+\sqrt{2D} W_t. \end{align}
What are the fundamental 1) modelling differences 2) mathematical differences between these equations?
Edit : I was a bit confused by this question and answer since it says we should NOT interpret this as sending the friction to $\infty$: Average velocity of overdamped particles in external field