I can reproduce (sort of):
Search on the main site: 1,449 results
Searching with the API: 857 results
I use a different filter which shows the total number of results at the bottom, but that doesn't matter:
What happens here is explained in the API documentation, if you look closely:
Usage of /search/advanced GET
Searches a site for any ...
There's indeed no difference between order=asc and 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 ...