These words get to be used very differently - but you may end up thinking there isn't really that much difference at all.
Psychodynamic is a generic description of a theoretical stance: the belief that human beings have an unconscious, and that the unconscious is a powerful hidden force, that controls thoughts and actions.
Psychoanalytic refers to a particular methodology for exploring the power of the unconscious; and it also refers to the body of theory that has been accumulated by clinicians who use this methodology.
This exploration traditionally involves an ongoing conversation, carried on at least once a week, and sometimes several times a week, between a clinician and a client. Through this conversation, the clinician helps the client to arrive at some understanding of his or her inner world.
The clinician is expected to join with the client in exploring the client's inner world.
Crucial to this process is the training that has been undertaken by the clinician. It is a lengthy and intensive training - as it needs to be, if the clinician is to have some hope of being able to lead an exploration of the unconscious communication that takes place between any two human beings, without contaminating the process by flooding it with too much of his or her own personal agenda. But every clinician is also a human being, so inevitably, something of the personal will seep into the discourse. What matters is that the clinician is always sufficiently self-aware to be exploring their own contribution to the conversation every bit as carefully as he or she explores the internal world of the client.
You may be wondering about the differences between 'psychoanalysis', 'psychoanalytic psychotherapy' and 'psychoanalytic counselling'. These terms usually describe differences in the length and intensity of training undertaken by the clinician. But it's always worth remembering that most clinicians continue to learn throughout their professional life, and someone who has had twenty years of experience and further training after a relatively short initial training may have acquired a great deal more understanding than someone who has just finished a lengthy and intensive training, but has had relatively little experience.
A psychoanalytic training requires the student to undertake their own analysis five times a week, and to see their patients five times a week under the supervision of a very senior clinician.
Clinicians offering psychoanalytic psychotherapy may or may not have had as intensive an experience of their own psychotherapy, and will see patients less frequently - maybe once or twice a week, or sometimes even less.
Both psychoanalysis and psychoanalytic psychotherapy base their work on the use of the counter-transference.
Both psychoanalysis and psychoanalytic psychotherapy use an ongoing conversation to build a relationship between the client and the clinician in order to explore the internal world of the client. Through careful observation of how the client relates to the clinician, they can collaborate in understanding together the way that the client tends to relate to the world at large.
Psychoanalytic counselling is a little different. It offers a conversation about an external problem. However, if the counsellor is trained to think psychoanalytically, he or she can use their counter-transference to understand the client a little better, and bring greater insight to the conversation. In practice, many psychotherapists use counselling to help their clients from time to time when external events are brought to them, and many counsellors who are trained to think psychoanalytically will provide psychotherapy to clients who want to look more deeply at their problems.