Classic Based Authentication :
- You cannot configure the Forms based authentication if your web application is using Classic Mode Authentication.You can convert a web application from Classic Mode Authentication to Claims Based Authentication. However, that can only be done using PowerShell commands and its an irreversible process.
- Classic authentication supports authentication types like Kerberos, NTLM, anonymous.
- Classic is more commonly seen in 2007 environments.
Claim Based Authentication:
- It enables authentication from windows as well as non-windows based systems. This also provides the capability to have multiple authentication in a single URL.
- Claims based authentication uses claims identities against a against a trusted identity provider.
- Claims are the recommended path for new deployments in SharePoint 2010