  This work presents new $^{12}$C + $^{12}$C reaction rates in the form of
numerical tables with associated uncertainty estimation, as well as analytical
formulae that can be directly implemented into stellar evolution codes. This
article further describes the impact of these new rates on C-burning in stars.
We determine reaction rates for two cross-section extrapolation models: one
based on the fusion-hindrance phenomenon, and the other on fusion-hindrance
plus a resonance, and compare our results to previous data. Using the GENEC
stellar evolution code, we study how these new rates impact the C-burning
phases in two sets of stellar models for stars with 12 M$_{\odot}$ and 25
M$_{\odot}$ initial masses chosen to be highly representative of the diversity
of massive stars. The effective temperatures of C-burning in both sets of
stellar models are entirely covered by the sensitivity of the present
experimental data, and no extrapolation of the rates is required. Although, the
rates may differ by more than an order of magnitude for temperatures typical of
C-burning, the impacts on the stellar structures during that phase remain
modest. This is a consequence of the readjustment of the stellar structure to a
change of nuclear reaction rate for reactions important for energy production.
For the hindrance case, the C-burning phase is found to occur at central
temperatures 10\% higher than with the hindrance plus resonance rate. Its
C-burning lifetime is reduced by a factor of two. This model, nevertheless,
loses more entropy than the other one thus enters earlier into the degeneracy
regime which will impact the last stages of the evolution at the pre-core
collapse time. The hindrance model produces up to 60% more neon. The impact of
the different rates on the s-process occurring during the C-burning phase is
modest, changing final abundances of s-processed elements by at most 20%
(cobalt).
