For binary classification, which is best Random Forest or Neural networks?
There is no best model for binary classification. Instead it is better to focus on what kind of data you have and let that guide your choice of model. 

Decision tree based models usually perform very well when dealing with regular tabular data of categorical and numerical features. Mainly random forest and gradient boosting (LightGBM and XGBoost). If you have very few samples a SVM can also be a good option.

Neural networks shine on more unstructured data like images and text. Time series are a bit more nuanced and often multiple models can work well.