Imperfect subgraph isomorphism
Your problem is Maximum Common Edge Subgraph Problem (Max CES) defined as follows:
given two graphs $G$ and $G'$, find a graph $H$ with the maximum number of edges
that is isomorphic to a subgraph of $G$ and to a subgraph of $G'$.

Proof: You are finding a subgraph $H$ of $G$ isomorphic to a subgraph of $G'$,
where $|E_{G}| - |E_{H}|$ is minimized. Since $|E_{G}|$ is an invariant of $G$,
$|E_{G}| - |E_{H}|$ is minimized if and only if $|E_{H}|$ is maximized.
Clearly, $H$ is isomorphic to a subgraph of $G$ and to a subgraph of $G'$. Q.E.D.

Approximability. In the Ph.D thesis of Kann, I found the description
"not known to be approximable within a constant" [3] (p. 115).
In a recent paper by Bahiense et al. [1],
it is mentioned that if $|V_{G}|$ and $|V_{G'}|$ are not required to be equal,
the problem becomes APX-hard.
But the citation for this result is an unpublished private communication [2].


L. Bahiense, G. Manic, B. Piva, C.C. de Souza. The maximum common edge subgraph problem: A polyhedral investigation. Discrete Applied Mathematics, to appear. doi:10.1016/j.dam.2012.01.026
M.M. Halldorsson, Personal communication, unpublished manuscript, 1994.
V. Kann. On the Approximability of NP-complete Optimization Problems. Ph.D. Thesis, NADA report TRITA-NA-9206, 1992. http://www.nada.kth.se/~viggo/papers/phdthesis.pdf