First, OneDrive and OneDrive for Business are not exactly the same. The former is a hosting service for your files – documents, videos, pictures, etc. OneDrive for Business app is essentially a tool that synchronizes your document libraries with your computer, so you can access them even offline.
OneDrive for Business on Mac
Just to give you a bit of background: OneDrive for Business was originally known as Groove. This tool was introduced a while ago, when Office 2007 was released. It was designed mainly to allow you to collaborate with others, i.e. work on documents together, without a server. You could sync files offline on all devices and continue working from home along with your colleagues. The application never became popular and by the time Office 2010 was released, Microsoft renamed it as SharePoint Workspace and redesigned it a bit. The software was transformed into a SharePoint tool to synchronize libraries offline. You probably know that the competition between SharePoint and Dropbox was pretty tough then.
With SharePoint 2013 release SharePoint Workspace has turned into SkyDrive Pro, its desktop installation enabled syncing SharePoint Document Libraries offline. “SkyDrive Pro” got Microsoft into a legal trouble and the company renamed the tool into OneDrive for Business.