Notts County FC is the official name. So the abbreviation 'Notts' is specifically associated with them.
Nottingham Forrest FC is the official name name of their rivals. So you'd be making a faux pas by calling them 'Notts'.
It would be like calling an Apple Mac a PC just because it's a personal computer. I mean, it kind of is. But you'd have to be out-of-the-loop and a bit naive to not understand the difference.
You'd get the same weird looks if you called Birmingham the Hammers or if you assumed that United referred to Newcastle United.