What are the typical differences between doing a math phd and an engineering phd on the same research topic?
There is a wide spectrum of possibilities in doing research in either mathematics or engineering. There are definitely people doing research under the auspices of an engineering department that is nearly as mathematically rigorous as in a pure math program.

The only exception to this is that pure mathematics programs do not look highly upon a fully applications-driven project under its auspices: there needs to be some sort of additional theory or analysis development.

But it's certainly possible to do a very highly mathematical project in an engineer department. My own thesis work amounted to algorithm development and analysis, with very limited application demonstrations. While I didn't have formal proofs the way you might in a pure math thesis, the project wouldn't really look like an "engineering" project—and certainly wasn't published in engineering journals!

So, in short, the main distinction I'd make is that one might expect to see more formal proofs and analysis, but no real distinctions in the kinds of problems that are tackled.