I sometimes meet IT service managers that are sure they know the true answer, but they become confused when they stay in the middle of a hurricane. It is easy when it’s written on paper, but when it comes to the real life – who could tell you, when an incident could cause a problem (or vice versa)?
TBH, it is easy:
An incident is just an unplanned service interruption. Fixed when user can proceed working with the system even if it requires workaround to be used. It is handled by a support team member and managed by support team supervisor.
A problem is a cause or a potential cause of one/many incidents. It might be created following an incident, but it’s always better to be proactive and prevent incident rather than fix them. It is usually handled by a group of persons in charge of particular service (sometimes including vendor/developers’ team) and managed by a service owners (IT and business ones).
No matter which software you use to automate ITSM, it is always about making IT service support processes simple and understandable by all the people involved, and different processes need to have different stakeholders.