Just to add a little more to the differences:
At the SP's side target realm where you will be federating to, is protected by different authentication schemes based on the federation configuration model as below.
1. Legacy Federation Model
* The TARGET realm must be protected by the SAML Authentication Scheme.
2. Partnership Federation Model
* The TARGET realm is protected by any regular authentication schemes.
If you look at the legacy model, the target realm must be protected by saml authentication scheme so in case if you wanted non-federated users to access, you would have a problem.
Non-federated users could SSO to the TARGET realm after logging onto other SP's realms but cannot directly logon to the TARGET.
On the other hand, Partnership Federation Model does not have this requirement and you are to protect the TARGET realm with normal authentication schemes.
This allows both federated and non-federated users to access and also get challenged if the user did not have a valid session.
Now here is a more realistic difference.
For Partnership Federation Model, there can be some creativity to it by protecting the SP's target realm using HTML Forms authentication scheme and setting the authentication scheme target to SP's AuthnRequest or IDP's Unsolicited service url to automate the federation.
That way, all users will be redirected to initiate federation in case if the user did not have a valid session.
==> this is not possible for Legacy Federation Model. You must have a link to initiate federation.