A problem, in an IT service management (ITSM) context, is an issue that could cause an incident. An incident, in turn, is an event that must be dealt with – or at least addressed with a workaround -- to ensure that the system it has affected, or could potentially affect, can continue to operate normally. Incidents typically involve the failure of an expected service.
The terms problem and incident are often confused. However, in ITSM, the two have a causal relationship: A problem is something that is wrong (or goes wrong), and an incident is the potential result of that problem. In security management, for example, a vulnerability such as an open port, is a problem, and something that indicates an attack has taken place as a result of that open port, such as a system crash, is an incident.
A problem can exist for some time without causing an incident but a single problem can also cause multiple different incidents. Likewise, a single incident may result from a combination of problems that would not separately cause an incident. Incidents may be resolved, meaning that service is restored, without discovering the problem responsible. However, without detecting the root source of an incident, there is no assurance that it will not recur.
See an introductory video clarifying the difference between a problem and an incident in ITSM: