The elisp manual has a section for defining menus: https://www.gnu.org/software/emacs/manual/html_node/elisp/Defining-Menus.html. It says,
A keymap acts as a menu if it has an overall prompt string [...]
Emacs displays the overall prompt string as the menu title in some cases, depending on the toolkit (if any) used for displaying menus
[the string] is required for menus which do not use a toolkit, e.g., on a text terminal.
So, there's your answer. And what that means in practice is that you're probably better off using the same string for both, so that your emacs interface is as similar as possible in terminal and GUI mode.
As a side note...
(define-minor-mode dilna-mode "docstring"
:keymap (make-sparse-keymap "Dilna Mode")
:lighter " DNA")
(easy-menu-define dilna-menu dilna-mode-map "docstring"
'("Easy Dilna Menu"
["Emacs Version" emacs-version t]))
(easy-menu-define dilna-submenu nil "docstring"
'("Dilna Submenu"
["Emacs Submenu Version" emacs-version t]))
(easy-menu-add-item dilna-menu nil dilna-submenu)
This adds a GUI menu called "Easy Dilna Menu". The lighter menu has a "title" of "Easy Dilna Menu". Both menus have a command to show the emacs version and a submenu with a single command to also show the emacs version.
It's worth noting that this is very close to Getting easy-menu to display a submenu and that the documentation is really very clear about getting this.
You're probably better off using easy-menu-define
.