I'd like to know which of this two expression is grammatically correct to show in a message in a website, when the users filter and there's no data returned from the server.
One difference I can think of (though both variants are indeed correct, as has already been commented) is related to whether your message appears during or after, say, the search.
"No answers were found" is quite explicit as to what is meant:
A search was done, but it yielded no results.
"No answers found", on the other hand, loose as it is, may connote that the search didn't yield and is still not yielding any results, so it may be more suitable in case the message appears in real-time, while the search is still being executed.