Let me add a very concrete application of higher-dimensional differential geometry (though maybe you know about this one already @dlfivefifty): The complete description of electricity and magnetism is given by the Maxwell equations. These are greatly simplified by using 4-dimensional differential geometry.
Instead of considering the electric and the magnetic field as two separate 3-dimensional entities, you consider them a single 2-form in 4-dimensional space. It then reduces to a single equation and, even better, this immediately explains why they are so closely intertwined and why what’s an electric effect to one observer can be a magnetic effect to another.
This is very concrete – it’s the kind of physics that makes GPS tick.