Plan Nickname (Internal) is the required name used inside the admin: it is what you see in the plans list, in the Current Subscriptions tables, in the Price Change Log and in reports.

Plan Name (Shown to Customers) is the label printed on the plan button of the customer subscription page. If you leave it empty, Sprwt falls back to the internal nickname, so you only need it when you want a different public name (for example "Lean & Clean" internally vs "Lean & Clean – 5 day" publicly).