Also, training a model is not as easy as dumping data in. NN's often suffer from high variance, so you need to constantly make slight adjustments. This cycle of adjust-process-analyze is very time-consuming both in terms of computing time (even on Google's servers training can take a few hours) and human-time.
Sometimes you'll let lucky. You can build a NN that gets accuracy in the ~80% range for handwriting recognition with ~50 lines of code, if not fewer. But that missing 20% is critical for any important task and getting there requires a lot of "parenting". And most times you won't be working with a vanilla NN and you won't be getting more than ~50% to start with.
It's also important to note that NN's are not a panacea; in fact, they're often not the right tool for the job. They tend to be outperformed by simple statistical learning techniques in a variety of tasks. Deep NN's can do a lot, but require a lot of data and constant adjustment of hyperparameters.
The biggest advances and most impressive predictions these days come from a combination of techniques and models, and these ensemble methods require a lot of work on part of us humans. Ensemble learning is where the magic really happens, and by magic I mean tons of work.