I think barge is more forgiving but the weldwood holds better when right. I've found with weldwood to put it together when it's almost still too wet and pull back apart, then when you stick it the next time it will literally tear the leather trying to get it apart.
I always use the barge on large areas like gluing a lining for that reason, it's just easier and more forgiving. For a structural bond I like the weldwood.