Is there anything that non-newtonian calculus can do, which newtonian calculus cannot? As well, are there problems where non-newtonian calculus leads to a more elegant or simple solution than regular calculus?

I think that there is nothing that non-Newtonian calculus can do that cannot be done with Newtonian calculus. The reason is that every non-Newtonian derivative or integral can be expressed using Newtonian derivatives or integrals respectively. That said, I think there are problems where non-Newtonian calculus leads to a more elegant or simpler solution. A concrete example would be the well known Cole Hopf transformation that converts a nonlinear PDE into the linear heat equation. The original non-linear PDE is only non-linear using Newtonian derivatives for the spatial variable. It is linear when using the so-called geometric derivatives for the spatial variable instead. I can give other examples.