Polynomial regression vs. multilayer perceptron
Polynomial regression can have multiple entries in the normal equation and it is not easy to say which polynomials you have to use in advance. Moreover, if you have lots of features you cannot handle memory errors most of the time. Nowadays people use MLPs and use batch normalization among layers for learning better. Those that you are referring to are a bit old algorithms but the former one is the logical mathematical solution for learning problems and the latter one is a beginning point for deep neural networks. I recommend taking a look at here and here.