The difference is context.

A neighborhoodisan open set, but when we use the term neighborhood, we usually are referring to the set of points closer than a certain radius around a specific point.

When we talk about open sets, we are talking about a set where every point has at least one neighborhood that is completely contained in the set (i.e. it doesn't necessarily have a metric "circular" shape like a neighborhood.). Also, some open sets can be the union of two separated open sets, but neighborhoods cannot be.