Yes – the index table is used even for queries with quotes. Words in the quotes are treated as singular keywords, from search perspective there is almost no difference.
Actually, I mean when I search a keyword, say “John” (without quotes) and hit Enter, the speed of search results appearing is rather slow compared to the ajax results. Ajax is like 3 seconds but search results on the page are like 30 seconds or more.
It will execute the same query for both the normal and the results pages. The live search is usually a bit faster, as WordPress does not have to load up all the front-end resorces, scripts, styles, most hooks etc..
The search query is the same, if there is slowness, then it’s more likely due to an overhead. If you want I can debug it in place via FTP access though.