ANSWER: Strictly speaking, none. The terms errors and omissions insurance and professional liability insurance usually refer to the same type of policy. Of course, you may find that one insurer chooses to use one of the two terms while another opts for the other and that the extent of the coverage differs from one to the other.
However, both policies will essentially protect you against the cost of compensation claims when a client believes you made a mistake or were negligent in the conduct of your business, leading to damage or losses.