What are advantages of Artificial Neural Networks over Support Vector Machines?

Judging from the examples you provide, I'm assuming that by ANNs, you mean multilayer feed-forward networks (FF nets for short), such as multilayer perceptrons, because those are in direct competition with SVMs.

One specific benefit that these models have over SVMs is that their size is fixed: they are parametric models, while SVMs are non-parametric. That is, in an ANN you have a bunch of hidden layers with sizes h₁ through h_n depending on the number of features, plus bias parameters, and those make up your model. By contrast, an SVM (at least a kernelized one) consists of a set of support vectors, selected from the training set, with a weight for each. In the worst case, the number of support vectors is exactly the number of training samples (though that mainly occurs with small training sets or in degenerate cases) and in general its model size scales linearly. In natural language processing, SVM classifiers with tens of thousands of support vectors, each having hundreds of thousands of features, is not unheard of.

Also, online training of FF nets is very simple compared to online SVM fitting, and predicting can be quite a bit faster.

EDIT: all of the above pertains to the general case of kernelized SVMs. Linear SVM are a special case in that they are parametric and allow online learning with simple algorithms such as stochastic gradient descent.

One obvious advantage of artificial neural networks over support vector machines is that artificial neural networks may have any number of outputs, while support vector machines have only one. The most direct way to create an n-ary classifier with support vector machines is to create n support vector machines and train each of them one by one. On the other hand, an n-ary classifier with neural networks can be trained in one go. Additionally, the neural network will make more sense because it is one whole, whereas the support vector machines are isolated systems. This is especially useful if the outputs are inter-related.

For example, if the goal was to classify hand-written digits, ten support vector machines would do. Each support vector machine would recognize exactly one digit, and fail to recognize all others. Since each handwritten digit cannot be meant to hold more information than just its class, it makes no sense to try to solve this with an artificial neural network.

However, suppose the goal was to model a person's hormone balance (for several hormones) as a function of easily measured physiological factors such as time since last meal, heart rate, etc ... Since these factors are all inter-related, artificial neural network regression makes more sense than support vector machine regression.