Which bond - C-C or C-H - is stronger?
The "strength" of a bond can't be evaluated just based on its enthalpy.  

According to this relation

$$ \Delta H = \sum \Delta H_\text{bonds broken} - \sum \Delta H_\text{bonds created} $$
the strength of your bond will depend on the reactive nearby the bond. If the molecule is likely to create bonds with higher enthalpy after the bond's breakdown, then the enthalpy variation will be negative. The process will then be spontaneous and the initial bond will breaks "easily".
There is also some steric consideration, the hydrogen atom is smaller, the bond is then more easy to be "attacked".
Finally, for a polar bond, the electronic cloud is delocalised closer to one atom rather than the other (according to their electronegativity) so you have "less perturbation to add" to fully remove the covalent electron from the less electronegative atom.