近所 is the everyday ordinary word for a part of a town where people live, a collection of residential buildings, the most basic meaning of neighborhood.
Generally speaking 付近 and 近隣 are used in situations where “neighborhood” might be possible, but it’s more like “vicinity” or “adjacent area”. It’s not usually used for talking about residential neighborhoods.
付近 and 近隣 are also more likely to be seen in writing and on signage than in conversation.
For instance
ドアの付近をあけて、 ドアにもたれないでください Leave the area around the door open and do not lean on it. (as you might see on a train)
東京付近の水族館 aquariums in the areas around Tokyo
近隣のホールに入らないでください Do not enter the neighboring hole (as you might see written on a golf course near where the people from another hole might be hitting their tee shots)
There is yet another word for neighborhood, 近傍, which is used in mathematics (in the context of topology). Do you also see this used in everyday life, or is it a more obscure one?