Late in the day, but:
The concave bow allows a lighter bow to be used for the same level of tension in the hair, because it reduces the torque over the majority of the stick's length (and particularly at the centre where the effect of a bend is greatest). Greater tension means that, for a given change in force, you have to move the bow less far, which is useful in loud-and-fast passages – less-weight-and-more-tension is a double-whammy for high-speed changes of bow pressure.
There are costs, of course: for example, you cannot afford to reduce the tension to play multiple strings simultaneously (the technique of tightening the bow with thumb or forefinger is no longer relevant).
BTW, I don't think the reduction of rotational inertia around the length of the stick is an advantage – as you really don't want to rotate the stick in that direction in the first instance.
N.B.1: that the improved stiffness was already well understood and implemented before Tourte. Indeed, late baroque and very early classical bows included sticks where, before tightening, the stick was almost touching the hair in the upper half if the bow. I believe that the advantage of the Tourte is supposed to be increased flexibility when playing at the tip, but I could be entirely wrong on this.
N.B.2: I personally prefer a slightly short, stiff concave snakewood bow for most 17th-century and earlier music. Indeed, even for late romantic and modern works, I find it preferable to some brazilwood bows supposedly built to Tourte's design, but I'm probably somewhat atypical in this.