From what I can see the front page is showing the default search, not the Ajax Search Pro, so it's definitely not a styling issue. This might be a programaticall error. Make sure that the same header file is used on the front page as on the other pages. It seems to me as if the front page was displaying the default search box and others the search shortcode (either in an if-else statement, or a different file is used for header on the front page).
If that's all right, than this must be a simple page cache issue. One of the layers, either W3TC, Cloudlare or MaxCDN might be serving an older cached version of the front page, which doesn't use the search shortcode yet. Make sure to clear the page cache, browser cache, Cloudflare and MaxCDN as well.
Also, test the front page if W3TC is disabled. If ajax search pro is visible when W3TC is inactive, then it's only a page cache showing, and it's not cleared. If still the old box is showing, well then there is somewhere a programatical error, displaying the old box instead of ajax search pro shortcode.Best,
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