BOSH is...
a transport protocol that emulates the semantics of a long-lived, bidirectional TCP connection between two entities (such as a client and a server) by efficiently using multiple synchronous HTTP request/response pairs without requiring the use of frequent polling or chunked responses.
This sounds like WebSockets and HTTP long-polling except that it uses two open HTTP connections instead of one and doesn't extend the HTTP protocol.
What are the differences between the two protocols, and what use case would prefer WebSockets over BOSH?
First let me address WebSockets readiness:
WebSockets implementation of the Hixie-76 protocol are shipped and enabled by default in Chrome, Safari and iOS (iPhone and iPad). The Hixie-76 protocol is also shipped but disabled by default in Firefox 4 and Opera 11. The web-socket-js project is a Flash shim/polyfill that adds WebSocket (Hixie-76) support to any browser with Flash.
In other words, WebSockets is available for almost every browser in the wild.
The reason why Opera and Mozilla chose to disable the protocol by default is because of a theoretical concern that there might be some broken HTTP proxies/intermediaries that could be attacked/poisoned using the Hixie versions of the protocol. The same concern applies to Flash but Mozilla and Opera felt a higher duty of responsibility for code that they ship. The HyBi versions of the protocol (the protocol was moved to the IETF HyBi working group) address the security concern.
Mozilla, Opera, Google, and Microsoft are all working on HyBi protocol implementations (although Microsoft is maintaining theirs as a separate download for now). There is a branch of web-socket-js with HyBi-07 support.
Update: As of Feb, 2013, the latest HyBi/IETF RFC 6455 spec is supported by Chrome 14, Firefox 7, IE10, Opera 12.1, Safari 6.0 and by the web-socket-js Flash shim/polyfill. On mobile devices IETF6455 is supported by Safari on iOS 6.0, Opera Mobile 12.1, Chrome 14 for Android, Firefox 7 for Android, and Blackberry 7. The original default Android browser does not have any WebSocket support.
WebSocket servers are easy to implement. There are numerous standalone and plugin implementations most of which support both Hixie-76 and HyBi protocol versions:
BOSH vs WebSockets:
Update:
Just found a video where Ian Fette discusses the advantages of WebSockets over the Channel API which is similar to BOSH (at 44:00)