Disabling this option means that SharePoint will not crawl the items in the list and they will not come up in a general search, as you say.
In my experience, it does impact searching within the list, as well. The search box at the top of the list will still be visible, but if you enter a keyword and hit enter, you will receive the message "Your search returned no results."
Items in SharePoint need to be crawled for any search box to work. So, if you have disabled that functionality, the search box on the site and the search box at the top of the list will not return results from that list.
To hide the search box at the top of the list, edit the page, then edit the web part. In the web part properties menu, under miscellaneous, uncheck "Display search box." Click OK to save your changes on the web part and stop editing to save your changes on the page. The search box at the top of the list should now be hidden.