Happy to be corrected if I am wrong, but I have been spending a lot of time researching this and I am seeing some mixing up of terms.
The E3 is of a non-immigrant type, (read a -Temporary- Visa) and is only valid for something like 2yrs, which of course you can then renew. As such you get a Work Permit or called a Employment Authorization Document (EAD) which allows you to work. Think of it as a temporary permit that shows your "status" i.e. can you legally work.
The Green card’s main function is identifying its holder as a lawful permanent or conditional resident of the United States. If you have resident status then this implies you can legally work there.
A Visa allows you into the country for a specific reason, and with varied restrictions e.g. time, can/cannot work. Depending on the Visa this then allows a specific Status or EAD.
Depending on what criteria you meet for what visa, and what sponsorship you get dictates what you can apply for. I think people here are mixing up what the heck visa they or their 'friends' are getting. Personally I have the possibility of marrying my US girlfriend, but even that is no good because it looks like it takes nearly a year to get the K1 visa to enter US to do it. If I marry on a tourist visa I am up for Fraud.