Both of them are possibly based on puppets, with Shuppet being a hand puppet (specifically a teru teru bōzu) and Banette being a marionette. You can view this progression as an evolution from a simple, easy to control type of puppet to a more complex and articulated design. Alternatively, you can view Shuppet as a cloth/rag and Banette as a sewn doll, with the progression being that it was stitched up to have a solid form, zippers, and defined limbs.
It's common for people to swap out the eyes of dolls, especially old ones, so you could interpret the change in the eyes that way. Or you could imagine that Banette developed more frightening eyes due to the resentment mentioned in its Pokedex entries. Its grey body color could be due to fabric fading from age and neglect.
sumwun's link gives some interesting details about the Shinto beliefs that could have inspired these Pokemon, mainly pertaining to the idea of different objects being able to come to life if treated poorly. Since their themes are so similar, it would be odd if both designs were made only for them to be totally unrelated species, especially since Shuppet being so weak makes it pretty useless if it doesn't evolve.
But ultimately, there isn't really any special reason why Banette looks kind of different from Shuppet... Why not? Lots of Pokemon have had significantly different looks than the other Pokemon in their evolutionary line and it's been like that since the first generation. Shuppet and Banette represent a pretty tame change compared to how Pokemon like like Dratini, Magikarp, or Clamperl turn out. Pokemon evolution itself has been described as being most similar to metamorphosis or puberty—processes that can make a creature look totally different, not just bigger.
