They're not really all that close. For instance that Autograd library requires that you use its own version of numpy instead of the regular version because it can't differentiate C. Because most non-trivial python code is actually written in C, there's almost no performant programs you can just differentiate out of the box unless autograd itself has a fork of that package.
My understanding is that the python Autograd library is also quite slow (though it's been a while since I looked at it).
Zygote will give the same runtime performance as handwritten derivatives in many cases and works across almost the entire julia ecosystem since nearly all julia packages are truly written in julia.
JAX (github.com/google/jax), which is being developed by many of the authors of Autograd, is a probably a better comparison. At the cost of requiring you to rewrite control flow in a functional way, it eliminates Python's overhead by compiling NumPy code into XLA.
My understanding is that the python Autograd library is also quite slow (though it's been a while since I looked at it).
Zygote will give the same runtime performance as handwritten derivatives in many cases and works across almost the entire julia ecosystem since nearly all julia packages are truly written in julia.