While I can imagine that neural networks are useful in predicting rainfalls or precipitations, I don’t see how they are suitable for some tasks, such as the forecasting of typhoon paths. A neural network is basically a function interpolation device. To use it for prediction, the implicit assumption is that the function it approximates is somehow well behaved, but some weather systems are chaotic. Function interpolation doesn’t seem very useful in this regard.

From what I see from googling, cyclone tracks forecasting by neural network is definitely an active area of research, but apparently they aren’t practical yet. The lady you talked to were probably working on some chaotic systems like this.