This confused me for a while when I first learned it, so in case it helps anyone else:

The standard logistic sigmoid function, a.k.a. the inverse logit function, is

$$g\(x\) = \frac{e^x}{1 + e^x}$$

Its outputs range from 0 to 1, and are often interpreted as probabilities (in, say, logistic regression).

The tanh function, a.k.a. hyperbolic tangent function, is a rescaling of the logistic sigmoid, such that its outputs range from -1 to 1. (There’s horizontal stretching as well.)

$$tanh\(x\) = 2g\(2x\) - 1$$

It’s easy to show the above leads to the standard definition $tanh\(x\) = \frac{e^x - e^{-x}}{e^x + e^{-x}}$ . The (-1,+1) output range tends to be more convenient for neural networks, so tanh functions show up there a lot.

The two functions are plotted below. Blue is the logistic function, and red is tanh.