The whole point of a websocket connection is that you don’t ever have to ping the app for changes. Instead, the client just connects once and then the server can just directly send the client changes whenever they are available. The client never has to ask. The server just sends data when it’s available.
For any type of server-initiated-data, this is way more efficient with bandwidth than http polling. Besides giving you much more timely results (the results are delivered immediately rather than discovered by the client only on the next polling interval).
For pure bandwidth usage, the details would depend upon the exact circumstances. An http polling request has to set up a TCP connection and confirm that connection (even more data if its an SSL connection), then it has to send the http request, including any relevant cookies that belong to that host and including relevant headers and the GET URL. Then, the server has to send a response. And, most of the time all of this overhead of polling will be completely wasted bandwidth because there’s nothing new to report.
A webSocket starts with a simple http request, then upgrades the protocol to the webSocket protocol. The webSocket connection itself need not send any data at all until the server has something to send to the client in which case the server just sends the packet. Sending the data itself has far less overhead too. There are no cookies, no headers, etc… just the data. Even if you use some keep-alives on the webSocket, that amount of data is incredibly tiny compared to the overhead of an HTTP request.
So, how exactly much you would save in bandwidth depends upon the details of the circumstances. If it takes 50 polling requests before it finds any useful data, then every one of those http requests is entirely wasted compared to the webSocket scenario. The difference in bandwidth could be enormous.
You asked about an application that only needs periodic checks. As soon as you have a periodic check that results in no data being retrieved, that’s wasted bandwidth. That’s the whole idea of a webSocket. You consume no bandwidth (or close to no bandwidth) when there’s no data to send.