Connections between Instance Hiding and Fully Homomorphic Encryption
Here's what's common in both scenarios: Alice holds $x$ and wants to compute $f(x)$. A server Bob will help with the computation. Alice doesn't want Bob to learn $x$, so she sends some encoding of $x$ that hides $x$. Bob does some work and sends back a result from which Alice can infer $f(x)$.

The two settings differ on: (1) how Bob is allowed to help Alice with the computation, (2) what kind of guarantee Alice has about the privacy of her data $x$.

In the setting of the Abadi-Feigenbaum-Kilian paper, Bob must evaluate $f$ itself on the stuff that Alice sends to him. He is not allowed to compute some other thing related to $f$. This is quite a severe restriction! Also, this setting demands unconditional privacy against Bob.

In the setting of FHE, Bob doesn't literally perform the computation $f$ on what Alice sends him; rather, he performs the ``homomorphic evaluation of $f$ on a ciphertext.'' Also, Alice gets a privacy guarantee that is only computational.