One point to make is that no one embedding witnesses that a cardinal is strong (that is unless it is witnessing that $\kappa$ is something much stronger, like a supercompact).
The definition requires that for every ordinal $\alpha$ there is an embedding $j_\alpha$
with critical point $\kappa$ witnessing that $V_\alpha$ is the $V_\alpha$ of the target model.

I am trying to guess at what you may have seen: it is certainly true that if you
take the "$\kappa+1$"-extender $E$ derived from any such embedding $j_\alpha$ for $\alpha > \kappa +1$, then this
is essentially just a measure on ${\cal P}(\kappa)$, and then, yes $L[E] =L[U}$
by Kunen's analysis. 

To get an inner model for, e.g., a strong cardinal, one will need some methodology that allows you to build a class-sized predicate: one will need to have encoded somehow, for a proper class of $\alpha$,
those $j_\alpha$ on to the predicate. It is this methodology that makes the matter complicated; to get a fine structural inner model, such as $L[U]$ the construction is even more delicate.

The place to read about this is in Martin Zeman's book: "Inner Models and Large Cardinals"