The main advantages of using Artificial Neural Networks (ANN) include: it can handle large amount of data sets; it has the ability to implicitly detect complex nonlinear relationships between dependent and independent variables; it has ability to detect all possible interactions between predictor variables; etc.
You need to test the two algorithms under e.g., 10-fold cross-validation. If you found that they have almost same result, that means there is NO advantage, in your case, behind using Artificial Neural Network against regression techniques.
One of the advantages of ANN models is they are capable of learning from more than one input variables to determine the nonlinear relationship between input and output..