In exactly the same way that adding more terms to a polynomial fit causes more problems. The is one of the most fundamental results in the theory of statistical learning in general; don't blame Deep Learning for it.
Tradeoffs, tradeoffs everywhere. It's almost like traditional mathematical statistics has something to offer them fancy machine learners. (Breiman was a professor of statistics, after all... ahead of his time, but no less a statistician.)
More neurons means more parameters to adjust to your data, so overfitting is more likely to happen. It is like interpolation a function, the more parameters you use the more overfitting you have.
> if your data is somewhat different from your training data in what humans would call an insignificant way, your network may easily start to fail
humans call it insignificant because we have a deep knowledge of a lot of domains, meanwhile a network has been trained for an specific domain. So if you train the network with a distribution and then test it with another distribution it is not going to work. That is like quite obvious I think
Deep learning works incredible well. It works so well that it outperform humans in some domains. So may want to rethink what are you doing, because I think (but I may be wrong) the reason you are failing applying deep learning is something related with your process and not with deep learning
Yes, that's a problem right there, because often only one solution actually works, and you don't know which one unless you spend hours on training.