Machine Learning itself is kind of rebranding to avoid itself from being called yet another AI trick. It is AI as much as Deep Learning is. Main difference between traditional machine learning and deep learning is that, deep learning (neural networks) is inspired by biology.
They are closely related but entirely different approaches. Most machine learning is relatively simple statistical models. Deep learning means ridiculously large models. Sometimes they have millions of parameters and require rooms full of GPUs running for weeks to train. But the capacity means they can learn far more complicated functions (like machine vision or language.)
Machine learning isn't an approach, its an entire discipline. Deep learning is just a specific category of implementation of a subset (neural nets) of machine learning.
They aren't entirely different approaches, considering deep learning is a form of machine learning...
Real deep learning[0] is a very particular type of machine learning that has recently been shown to be quite useful for certain specific tasks. But as often happens with technical terms, it's been regularly abused once it started getting press.