Just a quick example of a graph that's an induced subgraph of a hypercube of dimension four, but not an isometric subgraph of a hypercube of any dimension.

Induced subgraph but not isometric subgraph of a 4-hypercube: an 8-vertex cycle, plus one more vertex adjacent to two vertices two steps apart from each other in the cycle, with vertices labeled to show the hypercube embedding

A more complicated but naturally arising example comes from the family of antimatroids on a set of three items, viewed as an induced subgraph of the powerset of the powerset (a hypercube of dimension eight).

A disconnected pair of vertices is an induced subgraph of a hypercube of dimension two, but not an isometric subgraph of any hypercube, but that seems too much a trick answer...