What is the purpose of the add_loss function in Keras?

I'll try to answer the original question of why model.add_loss() is being used instead of specifying a custom loss function to model.compile(loss=...).

All loss functions in Keras always take two parameters y_true and y_pred. Have a look at the definition of the various standard loss functions available in Keras, they all have these two parameters. They are the 'targets' (the Y variable in many textbooks) and the actual output of the model. Most standard loss functions can be written as an expression of these two tensors. But some more complex losses cannot be written in that way. For your VAE example this is the case because the loss function also depends on additional tensors, namely z_log_var and z_mean, which are not available to the loss functions. Using model.add_loss() has no such restriction and allows you to write much more complex losses that depend on many other tensors, but it has the inconvenience of being more dependent on the model, whereas the standard loss functions work with just any model.

(Note: The code proposed in other answers here are somewhat cheating in as much as they just use global variables to sneak in the additional required dependencies. This makes the loss function not a true function in the mathematical sense. I consider this to be much less clean code and I expect it to be more error-prone.)

JIH's answer is right of course but maybe it is useful to add:

model.add_loss() has no restrictions, but it also removes the comfort of using for example targets in the model.fit().

If you have a loss that depends on additional parameters of the model, of other models or external variables, you can still use a Keras type encapsulated loss function by having an encapsulating function where you pass all the additional parameters:

def loss_carrier(extra_param1, extra_param2):
    def loss(y_true, y_pred):
        #x = complicated math involving extra_param1, extraparam2, y_true, y_pred
        #remember to use tensor objects, so for example keras.sum, keras.square, keras.mean
        #also remember that if extra_param1, extra_maram2 are variable tensors instead of simple floats,
        #you need to have them defined as inputs=(main,extra_param1, extraparam2) in your keras.model instantiation.
        #and have them defind as keras.Input or tf.placeholder with the right shape.
        return x
    return loss

model.compile(optimizer='adam', loss=loss_carrier)

The trick is the last row where you return a function as Keras expects them with just two parameters y_true and y_pred.

Possibly looks more complicated than the model.add_loss version, but the loss stays modular.