The functions these two individuals perform are the same; there is no difference. I understand that both are of the same rank as well. Britain’s diplomatic missions in various parts of the world are either called “High Commission” or “Embassy”.
The former term is used to refer to Britain’s diplomatic missions in Commonwealth countries.
For example, the diplomatic mission in Delhi is called the High Commission, and the senior diplomat in charge of it is the “High Commissioner”.
Since the United States and Russia were never a part of the Commonwealth, Britain’s diplomatic missions in these countries are called “Embassies”, and not “High Commissions”. The person in charge of one is called “Ambassador”.