Given a two class multi dimensional classification problem, what reason would you give to choose Artificial Neural Network for carrying out the classification instead of Support Vector Machine or other classification methods?
SVM is Parametric. Parametric models are something with fixed finite number of parameters independent of dataset size. Anything which is not parametric model is non-parametric model. ANN is non parametric. Also ANN has 'deep architectures" which can represent "intelligent" behaviour/functions etc more efficiently than "shallow architectures" like SVMs. ANN may have any number of outputs, while support vector machines have only one.
I wouldn't give any reason to make an choice a priori, based on such a broad description.
Instead I would build models for NN and SVM (or any other that you were considering), train them on the data then compare using cross-validation and a suitable metric for your problem - such as accuracy, F1 score, AUROC or log loss. You may also want to note characteristics other than performance metric if you care about costs such as time to train, amount of computing resource required etc.
Experience may tell you likely best choices from the problem domain. For instance, if you are classifying photographic or scanned images, then currently a convolutional neural network would be a very strong candidate, as these have repeatedly shown themselves as best performers with real-world image data.