What I really noticed is that the Moby is bigger, definitely stretchier and softer. It's more of a "wrapped in a cozy blankie" feeling. The 'Bub feels firmer somehow ... kinda transitional between a Moby and, say, a Didy. I run hot, and I'm a smaller mama (5'1" - barely) so I have really loved my hug-a-bub. For the first year it was literally the only sling we owned. I can't use it for a back carry, tho.
HOWEVER at this point in my slinging career I would just buy a fabric I liked and hem the edges so the "tails" tapered. That 'bub was the single most expensive thing DD has got in her life so far. (it does come with a cool videotape tho ...)