Let's say I'm making a crumble topping or some biscuits. What, if any, differences would it make to the finished product if melted butter was used instead of rubbing it in to the flour? Thanks.
Melting the butter would create a more homogeneous mixture of butter and flour. This may sound like a good idea, but actually preserving little clumps of butter and flour in the final mixture for your crumble (or streusel, or pie crust, or US biscuit) is what gives the finished product its flaky texture. If you try using melting butter you will a lower-volume, denser, harder results which is undesirable in the applications you are describing.
You can even inadvertently get this result if you warm the butter up too much with your hands while creating the mixture; in such cases it can be helpful to return the dough to the fridge for a few minutes and letting the butter re-solidify before crumbling any large clumps apart again.