A neighbourhood can be a delineated area that might even have a name. There may be several recognised neighbourhoods within a town or city.
Is there a bank in the vicinity? Here, you would want to know if there is a bank near your current location, within a reasonably short distance. The answer might be "No, I'm afraid there's not" or "Yes, there's one just around the corner."
Is there a bank in this neighbourhood? Here, you are asking if there is a bank within a specific area of the town. The answer here would probably be "Yes, there are lots of them in the neighbourhood. Which bank do you need?"
Student or Learner