If we were to keep them in SciPy, they might belong in scipy.optimize next to check_grad and approx_frime. But I don't think these functions (as written) are very useful. They have obvious computational inefficiencies and very limited functionality. I would rather point users to a fully functioning library for finite-differences like findiff: https://github.com/maroba/findiff
I haven't used either of them, it just came up in a search for finite differences in Python.
Okay, thanks Stephan. Both look good, so unless someone has practical experience and can make a recommendation for why one of these is preferred, we should probably list both in the deprecation notice.