* Type/paste a URL
* Type/paste a search
* Search my browser history (usually to jump to a previous URL)
* Use search engine keywords to do direct searches on some applications I use regularly (eg, "jira P-123" does a search in JIRA directly, which happens to jump to that ticket directly)
Browsers that separate those two drive me a bit crazy, because of the extra thinking required before typing.
(I don't really like this whole "using the first search term as DNS lookup" but that's separate from the UX of single vs separate inputs.)