At the nanometer scale, the motion of a contact line separating a dry from a wet region is limited by the presence of surface heterogeneities that pin it. Here we revisit the seminal model proposed by Joanny and de Gennes to include the influence of thermal noise and viscosity using a Langevin model with two degrees of freedom: the average position of the contact line and its distortion. We identify the conditions under which the dynamics in a velocity-driven experiment can in fact be described by a constant forcing at small scale. We then relate the asymptotic properties of the relation between force and contact line velocity to the properties of the defects. In particular, we show that Kramers' approximation misses the strong asymmetry between advancing and receding directions. Finally, we show how to use the model to fit experimental data and extract the salient features of the surface energy landscape.