There is no 'spot' in your sentence! It is similar to 'place', but used in less contexts. It is typically used in 'a nice spot' or 'in one spot', meaning not moving.
'Over' always has an 'above' aspect. Weather is (almost) always above the ground, so you can refer to the weather being over a place. It is not very natural to use 'about' as a direct replacement for 'over' in that sentence. About is usually used to mean 'approximately' or to indicate 'moving around', which doesn't quite fit.