The Instant Find Next command doesn't have a button on the search toolbar because it doesn't use the search term you've entered on the search panel or any of the search options. It is listed in the Search menu, primarily so you can assign it a keyboard shortcut in the Keyboard Preferences. The default shortcut is Ctrl+Alt+ArrowDown, which is the most instant way of doing an Instant Find Next.
Instant Find Next looks for the next occurrence in the current file of the word under the text cursor. When the word is found, Instant Find Next selects it. If there is no word under the text cursor, or it doesn't occur again in the file, nothing happens at all. If some text is selected when you invoke Instant Find Next, and the selection does not span more than one line, Instant Find Next will search for the selected text rather than the word under the cursor.
The purpose of Instant Find Next and Search|Instant Find Previous is to quickly "walk" through the file, stopping at each spot where the word occurs. Since a successful match selects the word that was searched for, you can instantly search for further matches.
Search|Instant Find Previous