Roles-based authorization is used to group users into groups (roles) and then set permissions on the role rather than on individual users.
E.g: In your case you can create Admin role and provide permission to do "AddCompany, ViewCompany, DeleteCompany, EditCompany" tasks.
In this case easier to manage large set of users through small set of roles. This is the most commonly used model for authentication.
Claims-based authorization provides additional layers of abstraction on your authorization strategy. Further, claims are a method of providing information about an user rather than group of users. You create authorization policies that are used to generate a claim-set based on the authentication evidence presented by the user. Then the user presents claims to the application in order to access resources.
A claim is a statement that one subject makes about itself or another
subject. The statement can be about a name, identity, key, group,
privilege, or capability, for example. Claims are issued by a
provider, and they are given one or more values and then packaged in
security tokens that are issued by an issuer, commonly known as a
security token service (STS)
Resources : http://msdn.microsoft.com/en-gb/library/ff649821.aspx
http://msdn.microsoft.com/en-gb/library/ff649821.aspx
http://msdn.microsoft.com/en-gb/library/ff359101.aspx
Hope this helps.