Actuaries and accountants both work with the same information, both handle financial data, and both generate statistics. Yet each will perform different business functions, and will serve different purposes.
The majority of actuaries are employed in the insurance industry, and deal primarily with risk. They will provide the statistical probability of a future event occurring (such as accidents or natural disasters), and advise managers on how to reduce any likely financial impact of adverse events. They also advise insurance companies how much to charge in premiums and which customers to insure.
Accountants work with individuals or organizations, handling monetary transactions by recording financial information. Their job may also include financial analyzing and reporting, preparing tax returns, auditing accounts, and/or acting as consultants on a wide variety of financial matters. Their duties are typically broader than that of an actuary.
A neurologist is a medical doctor who specializes in treating diseases that affect the human nervous system.