Non-Semantic Search would see the query [Obama's Dog] and the two keywords in it; “Obama” and “Dog”. In this example it probably would return results containing the exact text “Obama’s Dog” but it could return results about Barack Obama or Dogs.
Semantic Search would understand the query [Obama's Dog] and immediately think “Bo”, then return results about the little mutt not results about Obama’s latest political news or what a dog is.
This is a poor example I know because enough material has been written about Bo to be returned in a non-semantic search but hopefully it explains the concept of Semantisicm and the future of web search.
Tim Berners-Lee, the guy responsible for the WWW called Semanticism the most important thing about Web 3.0 (The next stage of the web). It is artificial intelligence about relationships and comprehension of language, which is one thing computers in general can’t do at present. Semantic search is a part of that as search is part of our everyday lives. I’m trialling a semantic search engine at the moment and will get back to you with a full report.