There's indeed no difference between
order=desc; perhaps this is a limitation of Elasticsearch which powers the Stack Exchange search engine (both the API and on the sites). To be fair, to find the most irrelevant search result on the site you have to go to the last page yourself. The API is no different.
There is some difference between the API and the site, but have you seen this note on the documentation page (emphasis mine):
Searches a site for any questions which fit the given criteria.
Otherwise, the top results when searching for test with the API and on the site seem to match.
But if you want to mimic site search with the API, you're better off using the