I am just beginning to discover the numerous different types of equals within Common Lisp so I can't comment on them. I can probably comment on the C# ones however.
I think that by default '==' will default to the 'Equals' function. In C# however you can overload the '==' operator and make it work differently. An example of this might be an primary key comparison on objects stored in a database table. So you make '==' look at the key slot and return true or false based on this equality.
So I think the short answer is for the default behaviour they are exactly the same but you can overload the operator at which point they will be different.
Cheers,
Matthew