Balls are definable in metric spaces, neighborhoods in general topological ones, so "neighborhood" is indeed the more general term. That said, once we have a m

Balls are definable in metric spaces, neighborhoods in general topological ones, so "neighborhood" is indeed the more general term. That said, once we have a metric space, the metric balls give a natural basis for the topology, which means that every theorem about neighborhoods has a corresponding one that uses balls instead. So in practice, when in a metric space we just use balls everywhere because they let us exploit their extra structure if it's needed. This is why you see the abuse of language calling neighborhoods balls, despite that being technically incorrect.