  We study the fluctuations of the directed polymer in 1+1 dimensions in a
Gaussian random environment with a finite correlation length {\xi} and at
finite temperature. We address the correspondence between the geometrical
transverse fluctuations of the directed polymer, described by its roughness,
and the fluctuations of its free-energy, characterized by its two-point
correlator. Analytical arguments are provided in favor of a generic scaling law
between those quantities, at finite time, non-vanishing {\xi} and explicit
temperature dependence. Numerical results are in good agreement both for
simulations on the discrete directed polymer and on a continuous directed
polymer (with short-range correlated disorder). Applications to recent
experiments on liquid crystals are discussed.
