You can only really compare a webSocket connection to an HTTP connection if you measure something they have in common such as sending messages per second. If you are doing that, then a webSocket has a huge advantage in both bandwidth and in network traffic because it is an already established connection whereas an http connection must be re-established from scratch for every message and establishing that connection for every message has a cost.
Both a webSocket connection and an http connection have a bunch of overhead in order to set up the connection. There is, at a minimum, the TCP overhead of establishing a TCP connection which looks a bit like this:

Credit to this article
In addition a typical HTTP request will also include some overhead such as cookies and other headers.
The webSocket connection however, does this only once and then keeps the connection open so future messages can just be sent over that connection directly. The HTTP connection is temporal, it is connected, data sent, response received, then the connection is dropped. If you need to send another request, another HTTP connection must be established from scratch.
While the stateless nature of HTTP connections is great for conditions where a client only occasionally makes a request and often makes requests to different servers, it is NOT ideal if a given client is making lots of requests to the same host. In that case, it's much more efficient to establish one connection, keep that connection open and then just send messages over that existing connection when needed.
Given the general context of the article you mentioned and the other aspects it discusses, it appears that this difference is what that article is referring to when mentioning the difference in scalability.
Here's another related answer: Ajax vs Socket.io