I was going to chime in about neural nets too, since I thought that was what the article was going to mention instead of just defining it for logistic regressio

I was going to chime in about neural nets too, since I thought that was what the article was going to mention instead of just defining it for logistic regression. It is used because it is continuous and differentiable as you say, and it keeps the bounds neat between 0 and 1, although most modern deep neural nets use an ReLU, or rectifier function of some sort. The softplus is an ReLU, that has the nice property of its derivative being the sigmoid function.