Represent an octree as a binary tree of thrice the depth?
They're very similar, as you say.  Their asymptotic running time is equivalent; the difference in running time and space usage is at most a small constant.

However, in practice, constants can matter.  In practice, the octree might perform better, due to its better memory locality.  If the depth of the octree is $d$, the depth of your corresponding binary tree will be $3d$.  That means looking up an element requires three times as many pointer indirections in your binary tree.  Pointer indirections are expensive because they typically cause a miss in the cache (they cause random access to memory, which is the worst case for memory hierarchies).  Consequently, in practice, it's possible that the binary tree implementation might be up to three times slower than the octree, due to the increase in cache misses and the poorer memory locality.

Of course, the only way to know for sure is to implement and measure on some benchmarks.  But this is one plausible reason why octrees might be preferable in practice.