This is a great question - I would also love to know. I do Cathe's Barre Workout but honestly I don't like all of those "ballet-type" moves. I tend to have more fun with the weights.
Barre work is different than traditional weight work in that you usually do more dance/ballet specific exercises (without any assisted weights) and for very many repetitions (upward of 25 reps) per exercise. While these types of exercises generally promote muscular endurance over strength when you first start out you may initially develop a little more strength. But that transfers to endurance rather quickly because you generally dont use assisted weight with these exercises to make them more challenging.
These exercises are not better or worse then traditional weight lifting, they are simply different. As a seasoned exerciser, we need to constantly throw new exercises our way as well as change the exercise order of our existing workouts because our muscles adapt so quickly. Once a muscle has figured out where you are going with an exercise the results start to slow down because it no longer provides the total "shock" to the muscle for optimal muscle fiber recruitment.
So I suggest you use both methods of training and if you have one method that you like over the other, you can do that a little more than the other but be sure to change your exercise order within that workout around....this is where our premixes come in very handy