Recurrent neural networks work very well when working with sequential data, that is data where the order matters. In a regular fully connected network, the order of input doesn't really matter.
Another property of RNNs is that they can handle variable input and output. Consider again the simplified breast cancer dataset. If you have trained a regular FCNN on the dataset with the two features, it makes no sense to suddenly add a third feature. The network would not know what to do with it, and would reject such inputs with three features (or any other number of features that isn't two, for that matter).