Deep learning (and more generally machine learning, which includes approaches other than neural networks) is just one branch of artificial intelligence. However, it is far and away the single area that is experiencing the most explosive progress.
There is actually something of a debate among top experts about the role and importance of deep learning in AI going forward. Architects of Intelligence includes interviews with the pioneers of deep learning -- including Geoff Hinton, Yann LeCun, and Yoshua Bengio -- and they tend to believe that neural networks will ultimately prove capable of doing everything that really matters in AI. Other experts disagree. They think deep learning will have to be combined with other approaches in order to continue to drive the field forward.