From the perspective of how I work the answer is ‘nothing’ and there are many therapists who practice both but chose to label the work they do with either ‘counselling or ‘psychotherapy’, depending on personal preference.
The labels are often used interchangeably and there is a lot of overlap between the two if you explore each of them in more depth.
There is much debate in the industry about what the differences are ranging from counselling being a briefer form of therapy and psychotherapy longer-term to a psychotherapist requiring more training and experience than a counsellor. There is also a public perception that psychotherapy feels more clinical than counselling.
There are so many factors that make a counsellor or psychotherapist unique in their own practice that are beyond these labels – their training, experience, expertise, own life experiences, personality, theoretical approaches and models they use, on-going personal and professional development. What is most important is how you feel and what you think about your therapist and whether you feel able to work with them, to share your concerns, feelings, vulnerabilities and so on.
You may find the following article on the Counselling Directory website of further use. It also covers a brief definition of psychology and psychiatry.