ITIL Incident Management versus ITIL Problem Management

What is the differences between ITIL Incident Management and ITIL Problem Management, here is the explanation in simple comparison:

Process:
Incident Management: Deals with fighting symptoms to incidents

Problem Management: Seeks to remove the causes of incidents permanently from the IT infrastructure.

Interaction
Incident Management: interaction with customers is usually reactive, with the main objective being to find a workaround solution to restore normal services for the customer as soon as possible.

Problem Management: IT support staff are more proactive as they dedicate resources to establishing the underlying causes of incidents. There is usually little or no interaction with the customers, as this is left to the responsibility of the Service Desk.

Definition
Incident Management: Incident Management is to restore a normal service operation as quickly as possible, and to minimize the adverse impact on business operations, thus ensuring that the best possible levels of service quality and availability are maintained.

Problem Management: Problem Management investigates the underlying cause of incidents, and aims to prevent incidents of a similar nature from recurring. By removing errors, which often requires a structural change to the IT infrastructure in an organization, the number of incidents can be reduced over time.