Out of the box you can't specify different fields for the view/edit/new page layouts. You can only specify which fields are editable/required/read-only. Some options to consider.
Record Type Switching
Since the displayed page layout is based on the profile / record type you can in effect create two layouts one for a new record which is displayed when creating a new object, and another layout for viewing and editing an existing record. When a new record is saved use workflow or process builder to change the record type.
If custom development is an option you can override the new / edit / view pages for a case independently allowing you to display different fields in each scenario. This requires creating a visualforce page and/or lightning component.
With quick actions you can create a specific form for creating cases. It won't override the 'New' button globally, but you can hide that where possible and replace it with your quick action. Once the record is created the fields displayed when viewing or editing would be driven by the page layout.