When number of input features to be fed to the classifier is more it is preferable to go for support vector machines. But in the case of ANN as the number of features increases, the computational time taken increases drastically if the features are not disseminative.
In short whther to use SVM or ANN depends on dataset as well as no of input features.
If you use old (classic) NN, the answer depends on a lot of factors. But if you use modern NN - Convolution, multi-levels,...(i.e. Deep Learning), the answer is evident - NN!