The advantages and disadvantages will of course depend on the specific use case, but I'll try to point out some differences between WebSocket and HTTP.
WebSocket is more complex than HTTP. You can establish an HTTP connection with a telnet client, but you probably cannot do the same with WS. Even if you ignored the handshake requirements (which include the use of the SHA1 hash function), you would then be unable to properly mask and frame the data to be sent, and the server would close the connection.
As Uwe said, WebSocket connections are intended to be more persistent than HTTP connections. If you only want to receive an update every 30 minutes, you will want to go with HTTP. If you want to receive updates every second, a WebSocket might be a better option, because establishing an HTTP connection takes a lot of time.
To establish an HTTP connection, you first have to establish a TCP connection (SYN, SYN/ACK, ACK), then send a GET request with a pretty big header, then finally receive the server's response (along with another big header).
With an open WebSocket you simply receive the response (no request needed), and it comes with a much smaller header: from two bytes for small frames, up to 10 bytes for ridiculously large frames (in the gigabyte range).
You need to weigh the two costs (keeping a connection open vs establishing a new connection) to decide between the two protocols.
Note: this answer is based on the current draft of the protocol (draft-ietf-hybi-thewebsocketprotocol-09). WebSocket is evolving rapidly, many implementations are still based on older drafts, and some details may change before it is finalized.
