Neural network or other algorithms?
This is more of question how to select the correct machine learning algorithm, I would refer you to the following blog Which machine learning algorithm should I use?

Regression Algorithms models the relationship between variables that is 
iteratively refined using a measure of error in the predictions. Most popular examples are:


Ordinary Least Squares Regression (OLSR) 
Linear Regression 
Logistic Regression
etc  ...


On the other hand, Artificial Neural Networks models are inspired by the structure and/or function of biological neural networks. "Neural networks currently provide the best solutions to many problems in image recognition, speech recognition, and natural language processing." Neural Networks and Deep Learning/.  Neural Networks are hard to train; thus my recommentation not to start with Neutral Network.