Hi wroot,
We have two experiences here that we generally call the "Share Dialog" and the "Grant Access Dialog" and they serve related but slightly different purposes. We generally promote "Share" as the primary option as it works in the way many end users expect.
The Share Dialog helps the user get sharing links. These links convey access to people who use them. Links can have restrictions on them (ex. only works for people in your organization or only grants edit permission) and can be individually revoked. Links can be e-mailed directly from OneDrive & SharePoint or copied and sent via the app of your choice.
The Grant Access Dialog helps users grant permission to other users directly. Think of each item as having it's own list of people who are allowed to access it. Grant Access is all about putting the user into that list. Granting access offers some flexibility as well (permission level for example) and you can choose to notify the user (via an e-mail from OneDrive or SharePoint) or not.
Hope that helps!
Stephen Rice
OneDrive Program Manager II