Neural nets are from a silver bullet, and shouldn't be used where feature introspection is a huge requirement (this is why decision tree/random forest is popular), but they are far from being what they were in the 90s.
Note that I have a commercial interest in this so there's going to be inherent bias in my opinions.
To be fair, javascript isn't a scientific computing language. To do most neat training with neural nets, you're going to want to either scale them out, add more layers, and/or use GPUs. That being said, a neat toy example in javascript is convnetjs[1].
In fact, deep neural networks are trained in an unsupervised manner at first, but then back propagation is used to "fine tune" and improve the results. Because they require unlabeled data sets, and can perform so well, research into neural networks has experienced a recent resurgence.
By the way, any talk by Geoff Hinton is fantastic. If you are interested in neural networks and their capabilities, and you haven't already seen it, his Coursera course [1] builds from a simple linear perceptron to the current deep learning methods.
[1] https://class.coursera.org/neuralnets-2012-001 (You'll have to sign in to see it)
To summarize, people generally abandoned backpropigation trained neural networks for Support Vector Machines because neural nets require labeled and limited datasets, and work slowly and especially so when dealing with multiple layers which is sort of the whole point.
In my work in JavaScript, I was able to pull off only a single layer perceptron and it is neat but limited in what it can model.