Yes, i think that the French cup is more prestigious and popular by far. Because all the teams are involved, even the amateur one. Starting in july, with the little team of a 500 inhabitants town, to finish with the bigger 's ones : professionals teams.
Since few years, there are always big surprises, like a division 5 team beating a Ligue 1.
The League cup sounds more "commercial". A way to make more money for the professinal system.IMO