The short answer is that both B-fields are the same object!
The way the B-field comes to us from string theory it doesn't come alone, but comes together (among other fields) with the Riemannian metric. Due to the way both originate in the string, they are interrelated by what is called T-duality that mixes them and shows that both fields have to be regarded as two aspects of one and the same unified entity.
Physicists had understood various aspects of how these two fields unify, when Nigel Hitchin came along and noticed that there is a nice and useful mathemtical formalization of what is going on. This is the origin of generalized complex geometry.
But some aspects of the picture are still mising. For instance it is well-know that in full beauty the B-field is a gerbe with connection . Last I checked, this is represented in generalized complex geometry only rationally , meaning that the integral degree 3 class of this gerbe is seen only in its image in deRham cohomology.
This has to do with the fact that generalized complex geometry is really a theory of Couran algebroids (certain Lie 2-algebroids) and it is only their integration that knows about the full Lie 2-groupoids that yield the gerbe.
(I think I know the full story, but it is not written up yet.)