What is the difference between a logistic neuron in a neural net and a logit regression?

1. They both follow the Sigmoid function plotted below

Here is the R Code for reasons of reproducibility

f <- function(x) {
   1 / (1 + exp(-x))
}

curve(f, xlim = c(-6,6), ylim= c(-0.5,1.5), main = "Sigmoid function", col.main = "red")

2. They are both binary classifiers
3. Both classifiers are symmetric and give the same output for the same input.

But what is the difference between the two?