The `/search` path and the `/search/advanced` path **return only questions**, not answers, and [by design][1].  It says so right in [the doc pages][2]:

> Searches a site for any **questions** which fit the given criteria.  

and

> This method returns a list of **questions**.

Your example Stack Overflow search returned answers as well.  
The correct equivalent search is:

> [stackoverflow.com/search?q=raml+rest+is%3Aquestion][3]

-- which returns 6 results, the same as the API.

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The good news is that, as of version 2.2, the API provides a new route, `/search/excerpts`, that does return answers also.

If you [try it with your `raml rest` query][4], you'll see that it gives 18 results -- same as the Stack Overflow search.


  [1]: http://stackapps.com/a/655/7653
  [2]: https://api.stackexchange.com/docs/search
  [3]: http://stackoverflow.com/search?q=raml%20rest%20is%3Aquestion
  [4]: https://api.stackexchange.com/docs/excerpt-search#order=desc&sort=activity&q=raml%20rest&filter=!9YdnSO-*B&site=stackoverflow&run=true