One of the important features of the standard Sierpinski carpet as a fractal is the type of boundary between cells that is has, line segments or single points. One type of boundary that is ruled out by the local connectedness assumption is something like a Cantor set where if you took a neighborhood of a point on the boundary that neighborhood is not connected no matter how small it is. The point, so to speak, of Whyburn's characterization is that if any Cantor-like boundary occurred then the fractal is no longer homeomorphic to the standard carpet. Though it might still be an interesting fractal.
Side note, it is not too hard to make a fractal where the boundary between cells is a Cantor set. Take a three by three grid of squares and keep the top three and bottom three. This just the kind of example I mentioned.
I am sure there are other types of geometry that local connectedness precludes but this is an easy one to get your hands on.