Difference entity authentication and implicit key authentication
In entity authentication, the parties do not necessarily exchange a session key, but authenticate to each other by providing some proof related to their identites. For instance, if one party signs a challenge given by its partner, this party proves its identity. 

Implicit key authentication merely guarantees that the partner (with whom you are talking right now, but who could be unidentified) is holding the same key, and no one else is.

Example:


A chooses random $x$ and sends $g^x$ to B.
B chooses random $y$ and sends $g^y$ to A.


Only A and B can know $g^{ab}$, but they do not authenticate to each other. Hence, the protocol provides implicit key authentication but no entity authentication. The above protocol is actually not secure against man-in-the-middle attacks; it merely serves as an separation example.