The article details the difference between the Closed and Completed ticket statuses.
Kayako uses the 'Closed' status to denote conversations that are not only completed, but also will not be reopened. Once a conversation has been 'Closed' neither a customer nor an agent can open it again. Closed conversations also will not appear in views, but will appear in search results. We do this to make sure that your views stay snappy and responsive.
Unlike the Completed status, conversations are given the Closed status via automation. By default, Kayako comes with automation that will set conversations to Closed once it's been in the Completed state for five days.
You can edit this automation by going to Monitors in the admin area and editing the Close conversations 5 days after they've been Completed monitor.
NOTE: If you migrated from Kayako Classic, this monitor might not be defined.
Closed conversations are also archived periodically to preserve Kayako's performance and speed as your conversation archive grows. Kayako will automatically archive any conversations that are in Closed status for 60 days or more. When a conversation is archived, the user can search and view it, but cannot update it.