u can imagine them as 3 enclosed circles. the biggest one is the AI and inside it there are Machine learning circle and Deep Learning. Note that Deep learning circle is inside the Machine Learning one
AI is the broadest field which comprises of 'Machine Learning' that focuses on imparting the ability to the machines to learn from experience which may be through supervised, unsupervised or reinforcement learning. Now, learning methods could be shallow (consisting of 1, 2 or max. 3 layers of ANN training) or deep (consisting of >4 trainable layers).
Machine learning comprises different learning techniques, not only based on neural nets while "deep learning" is strictly about neural nets with special architectures like RNN, LSTM, MLP, CNN etc.
There are a few ways to look at this, but really I look at the applications of each of these topics, which makes it fairly obvious the differences, and also the benefits and drawbacks.
Machine learning is a bread term that is used for generating some sort of pattern from data. Typically these ML algorithms are used for classification or regression, such as being able to predict a stock price tomorrow based on some key factors, or being able to classify if a cell is cancerous based on protein expression levels. In these applications, the data is the key part. With a bunch of data, you can create a realistic and robust classification or regression algorithm. However, the key here is that a bunch of data is required, and YOU, the human behind the algorithm, come up with the predictors (in most cases). For instance, I want to look at how P/E ratios effect stock prices, or I want to look at the concentration of the protein levels from the MDM2 gene to attempt to identify a cancerous gene. YOU have come up with these ideas of what can predict and what wont.
Deep learning is really a subset of the above, it utilizes layers of transformations, such as convolutions in a CNN, to create features that, to the naked eye, make little sense. However, these features can be used to classify (or regress, in some special cases). Deep learning gets its name from the fact that there are layers of transformations of the original data to get the features that are then used to classify or regress, which is the goal overall of machine learning as we just discussed it. A typical example is a CNN that is fed images of pictures, when tells you what the item in the image is (google VGG16 or VGG19).
The final piece here is AI, which we COULD call any algorithm that learns from data, but recent progression in a field that wasn't mentioned in your original post, commonly known as reinforcement learning, is a bright and shining star in this field of AI. Remember how I said that YOU needed to decide the predictors in a ML algorithm? In RL, that is not the case. However, an RL algorithm doesn't just classify or regress. RL is used to MAKE DECISIONS. The fundamental idea is an agent-environment interface in which you have some sort of environment you're operating in, and you have some sort of state your agent is currently in, and you have to make some decision from that state to receive some reward from the environment. The idea here is to figure out an optimal policy for your agent to take at these decision points in order to maximize or minimize the total reward, such as maximizing profits in a trading day, or getting the most points possible in an Atari game. Its all about the DECISIONS made at particular STATES, and therefore is fundamentally based around a Markov Decision Process, and RL can really be considered a means of optimal policy estimation in the face of an MDP in which we don't know the model. There are traditional means of optimizing the policy in an MDP (if we know all the rules of the model), commonly known as dynamic programming, however RL takes it a step further and can answer that question if we have NO IDEA what sort of environment we're operating in.
If you're interested in the subject, there is a wealth of information on google and I would check out Barto and Sutton's book on RL.
I hope that clarifies some differences in the algorithms, as well as their possibilities and limitations.