One benefit that the ANN models have over SVMs is that their size is fixed: they are parametric models, while SVMs are non-parametric models.
Depending on the number of features, plus bias parameters, ANN has a bunch of hidden layers with sizes h1 to hn and those help in making up the 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. SVM has classifiers with tens of thousands of support vectors, each having hundreds of thousands of features. And those large number of features are not known to us thus no use of those features.
One more 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. Thus, ANN can train the models in one go and SVM has to train one by one.
Watch this video to learn about Neural Networks: