A website is set to be beta when it changed its interface and is in testing process. Websites, apps and software needs up gradation over a period of time.
When a website interface is changed several conflicts with screen sizes or loopholes in code may occur. To detect all such drawbacks in newer version its takes time. Instead of making a website, software or any app unavailable to users, it's beta version will be released.
Recently Alexa released its beta version which will be continued as permanent website in a few days. Also some websites and apps take feedback from testers on their newer version. If found any loopholes, they will be cleared or they'll continue with old interface.
There're no benefits except the user satisfaction for a new website. Actually a webmaster gets benefited with the website when his beta version passes the Browser compatibility issues, fits every resolution i.e. mobile, tablet, desktop and security issues etc.
I hope you got the solution.