The handrail is just the top part that’s the rail. The banister is the entire structure, including the Newell posts, the balusters (the smaller, vertical posts that run under under the handrail— which, together, are called a balustrade), and the handrail together.

It is customary to refer to the entire structure and to refer to it as the banister, unless you are speaking about a specific part of it. Honestly, the last time I used the word balustrade, it was to either talk specifically about the balusters I saw on a particular staircase, or I was talking to a contractor.

Diagram: (note: individual posts in a balustrade are known as balusters. The term balustrade refers to all of them together, supporting a rail.)