The basic distinction is between a certificate verifying control of a domain, and a certificate verifying the real-world entity behind the domain. With a standard SSL certificate, all that's verified is that the entity with the certificate legitimately controls that domain. It doesn't mean that that's the entity I think it is; I could register bankofamerica.co, and I can then legitimately get a domain-validated certificate for it, and that would show up as a green lock in browsers.
What that box indicates is that CAs have done more validation; EV certificates (the green box) generally require actually verifying the existence and name of the business requesting them. I could not get an EV certificate for that site that says "Bank of America" on it, because I don't have a company called Bank of America, and even if I did the actual person reviewing an EV cert application (unlike normal certs, EV certs aren't automated) would likely be somewhat suspicious at someone claiming to be a bank.
So that's the stated role of EV certs: Verifying that the server sending you a webpage is the correct server for that domain doesn't really help unless you also know that the domain is owned by the company you want to interact with. With Google and Facebook, you know already that their websites are google.com and facebook.com, so I know that I want to talk to google.com, and if I'm talking to the real google.com that's enough. With other organizations, it's not necessarily enough to know I'm talking to the real so-and-so.com; I also need to know so-and-so.com is the actual website I want to be talking to.