What I believe is the major difference is that with classic mode, you login directly, and if you need to be authenticated to a second server, it's not possible (you need to be logged in to this second server first -> new login screen). You will find that this is still often done but in that case the second connection is done under a different account (with privileges specifically for the resource, for instance).
With claims, you authenticate, you get a claims token and that is passed around to whoever needs it, and you are authenticated to this second server automatically without another login screen. Given that the server trusts your secure token service (whom initially authenticated you).
I may have the details wrong but in theory but this is generally how it works.
Claims and Kerberos are similar, but they differ in the technicalities.
Important part:
- Kerberos delegation — If the client authenticates with the front-end service by using Kerberos authentication, Kerberos delegation can be used to pass the client's identity to the back-end system.
- Claims — claims authentication allows the client's claims to be passed between services as long as there is trust between the two services and both are claims-aware.
- Claims authentication, like Kerberos authentication, can be used to delegate client credentials but requires the back-end application to be claims-aware.
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