Facebook allows two different types of "profiles" (accounts); personal and business. We all have personal profiles. The business profile is basically a stripped down version of a personal profile. The intent is for a business person who simply wants to manage their Facebook Page and/or ads without getting wrapped up in all the trappings of a personal profile. Business profiles do not appear in public search. They are really a sort of admin account for those with Pages and ads, because you need a profile to create a Page or to advertise on Facebook.
Facebook's TOS does not allow anybody to have more than one profile for any reason. Their logic is because their TOS also requires members to use their real name. If you can only use your real name to create a profile, then you can only have one account. Yes, many people are not using their real name and yes there are many personal profiles with business names. If/when they are caught or reported, they are deleted (sometimes given the option to rename) and yes, this does happen. If you create business profile, you can later upgrade it to a personal profile, but you can not downgrade a personal profile to a business profile.
Businesses must/should use Pages (the ones you used to Fan, but now Like) to list on Facebook. They appear in public search and can be accessed WITHOUT having a Facebook membership, amongst other features.
Facebook Groups are just for that...groups, clubs, fans, etc.
Example of Facebook "done by the book";
You decide you want to be on Facebook but don't want old classmates stalking you and such. You just want to promote your business. So you create a business profile (kirby ball) and then a Facebook Page for the business (Ball DJ and Lawn Mowing Service). Patrick sees some of your lawn work and thinks you are just the bees knees. He "Likes" your Page on Facebook. But that's not enough for him. He finds some others who also think you rock on a John Deer, so he creates a Group; I Like To Roll Around Naked on Ball's Lawns. (There is nothing preventing him from using the exact same name as your Page BTW) Later you become increasingly curious about Farmville and upgrade your business profile to a personal one. Nothing changes on your Page. However now others on Facebook can search, find, poke, poll and throw snowballs at Kirby Ball.
Any questions?