I am only casually acquainted with French, but took a course out of B&B. The two books are at approximately the same difficulty level. The primary difference is scope: French confines the discussion to mechanical vibrations and waves; B&B covers much of the material of French in chapters 1,2,7 and 8, but the rest of B&B is purely on electromagnetic waves. Check out the table of contents of these two books on Amazon and you will see chapters 3,4,5, and 6 of B&B covers aspects of electrodynamics - material not in French at all. B&B is a really good book, and chapters 3-8 are where it really shines. I am pretty sure there is no copying of French here. The main downsides of B&B are that you will have to get past the old fashioned typewriter type-setting, and that it has no solutions.
Since I have not seriously read French's book I cannot comment on whether it is better at explaining mechanical waves than B&B. Since MITs physics III (see ocw.mit.edu) uses both books, I suspect there is something about French that is either not in B&B or that is simply explained better.