What was the breakthrough?
Years later, Swiss researchers (Dan Ciresan et al) found that you can train neural nets with backprop, but you need lots of training time and lots of data. You can only achieve this by making use GPUs, otherwise it would take months.
Check out the publications by Ciresan on MNIST, have a look at Hinton's dropout paper or at the Kaggle competition that used deep nets. Or try it yourself and spend a descent amount of time on hyper parameter tuning. :)
It's been awhile since I read the paper, but as I recall it involved training an unsupervised model layer-by-layer (training a layer, freezing the weights, then training another layer on top of it).
The deep learning / RBM tutorial here is quite good and explains the technique.