They're not identical but they are related. There's a series of approximations and simplifications you can go through to get from biological neurons to neural nets. Essentially the weights in the neural net end up corresponding to steady-state firing rates of populations of spiking neurons. See for example Chapter 7 of Dayan & Abbott's Theoretical Neuroscience.
Not to be dismissive, but through any “series of approximations and simplifications” you can go from anything to anything else.
Discussing the right levels of abstraction is a huge thing in computational biology. At what level is 'the algorithm' of natural computation implemented?