Is 'deep learning' basically just neural networks with many, many hidden layers? Or is there something fundamentally different?

Is there a subtle difference between training a neural network, vs training a deep learner? Are different algorithms used?

As someone who's fairly comfortable with training reasonably small neural networks, how does one make a jump to deep learning? Is there even a jump?

(Sorry if that's a lot of questions bundles into one, but I was just trying to clarify my purposes)