You can understand the difference between these two concepts by reading below answer:-
The .save() you already have an object in your client-side code or had to retrieve the data from the server before you are writing it back, and you are writing back the whole thing.
On the other hand .update() does not require the data to be loaded to the client from the server. All of the interaction happens server-side without retrieving to the client. So .update() can be very efficient in this way when you are adding content to existing documents.