Neural Networks are a technique that allows software algorithms to appear to perform "intelligent" processing such as recognition tasks. This falls under the term "artificial intelligence".
Obviously there is more to AI than learning and NNs (or "deep learning").
Note that "artificial intelligence" never claimed to be something similar to human intelligence (or consciousness). Hence the term "artificial". It is just the study of systems that perform tasks that humans would regard as requiring some intelligence (even if none is needed. e.g. chatbots that blindly follow a simple set of rules or scrabble software that just brute force tries every possibility).