If you want to open a two-way connection between the browser and server, the only choice is to poll (hammer the server), or use comet (crufty and prone to disconnects).
Why don't browsers just let you open up a plain TCP connection? Is there any practical benefit to not having this ability?
Given that this functionality is effectively available through flash, there's no real security rationale - but these days no browser wants to be the first to implement a non-standard extension like that. Moreover, there's no easy way to do threads, which could make using a socket rather awkward.
The underlying protocol HTTP is basically a half duplex communication protocol which is stateless as well and does not supports full duplex communication. However, with HTML 5 websockets things are going to change. Websockets is a new standard which is being considered in HTML 5 specification. Once the specifications have been finalized and all the browser vendors have adapted the standards you can possibly use websockets to establish a dedicated TCP connection through browsers itself.
We must also keep in mind that HTTP was basically designed to deliver documents & share information between the geographically distributed teams and it was not intended to be a communication protocol as such.
Having said that, there are already companies which have built some messaging gateways to enable you to implement full duplex communication.
Over the years so many aspects or elements of the web have been hijacked in order to deliver richer experiences. Comet is but one example, where long lived connections were exploited in order to allow server side push. Originally web pages were just meant to be hyperlinked documnts of text and not the rich applications we often see today. Hacks and abuses of what the original thought intended will continue, until one day these things become more standardised.
The answer to your question is essentially no, there is no tangible advantage to not being able to open a two-way connection between client and server in a browser. The reason it can't be done is simply that this was not the intention of web browsers, which were developed to poll/retrieve documents. With the advent of Rich Internet Applications, it has become desirable to have such functionality, but previously this had never been the goal of a browser. Currently there is a void to be filled by an eventual protocol or implementation of an existing protocol which will govern two-way communication between a browser and the server. There are existing techniques used to simulate this behavior to different degrees (AJAX, Comet, etc.) or it can be accomplished with embedded objects (Java, Flash, ActiveX Controls in IE) but these are simply paths around the void, not bridges over it.
We will simply have to wait (or act) for the standard to be written and the implementation to follow. More than likely, the implementation will actually come first, and we will have a fistfull of new cross-browser compatibility issues to enjoy :) Oh, the bleeding edge!
Firewalls. Non-HTTP traffic is often blocked by firewalls, so opening up a random TCP port for communication will often fail.