Apart from the obvious push capabilities, yes, websocket has much less overhead and is faster than HTTPS.

Once the connection is established, you can send requests starting at 1 byte size, whereas HTTP needs the full headers on every request.

HTTPS also has a lot of handshake overhead, where you need 2-3 round trips per request instead of a single one with websocket. This can make a world of difference in a 150+ ping situation, where HTTP feels sluggish and websocket feels instant.