The choice of multi-class classification technique is only a matter of preference. If applied properly, both may produce comparable results in most cases. The accuracy may differ based on the application and the nature of features at hand.

My personal preference would be ANN, since it is highly flexible when considering multi-class classifications. In ANN, you wouldn't have to worry about kernel selection as in SVM.

A common way to do multi-class classification in SVM is using the one-vs-all method. Materials for which is scattered over the Internet. You may refer what best suits your understanding.

Thanks a lot.