近所 and 付近: neighborhood and vicinity
Compare a nearby neighborhood with the vicinity of a reference point.
Near where one lives
近所 (きんじょ) refers to a nearby neighborhood or area. 「近所の人に会いました」 means meeting someone from the neighborhood, not necessarily the next-door neighbor.
「近所にパン屋があります」 says that there is a bakery in the neighborhood. It does not specify a distance in meters.
Near a reference point
付近 (ふきん) means the vicinity of a particular point. 「駅付近で待っています」 says that the speaker is waiting near the station, not necessarily inside it.
「この付近に店はありますか」 asks whether there are shops nearby. It can refer to a place unrelated to the speaker's home or routine area.
No fixed distance threshold
The words can overlap for nearby areas, but 近所 suggests a neighborhood and 付近 the area around a point. Do not mechanically replace expressions such as 「近所の人」.
「付近を探しましたが、見つかりませんでした」 means searching the vicinity without finding the item. Distinguish the area searched from the outcome.
Does 「駅付近」 identify the exact exit for meeting?
Show the answer and why
No. It only indicates the station's vicinity. Confirm an exit, landmark, or other precise location.

