In Joel David Hamkins "Forcing and Large Cardinasls" a definition of $extender$ embeddings:

"An embedding $j:V \to M$ is an $extender$ embedding if every element of $M$ can be represented in the form $j(f)(\alpha)$ for some $f:\kappa \to V$ and $\alpha < j(\kappa)$, where $\kappa$ is the critical point of $j$."

Every ultra-power embedding by a measure $\mu$ on a measurable cardinal $\kappa$ is an extender embedding since $\kappa = [id]_{\mu}$ can be the seed for such an embedding and therefore generate $M$ by representing every element as a $j(f)(\kappa)$ for an $f:\kappa \to V$, $\kappa$ is, of course, the critical point.

However, not every extender embedding is an ultra-power embedding. For such embeddings j: M $\to$ N, how are the images of such embeddings different from ultra-power images? Or, how the embeddings are different? And, what mathematics can be done in an extender-non-measurable-ultra-power embeddings?