The character ~ before a word in a query will have Google search for both actual occurrences of the word plus synonyms.
For the end user - the search engine visitor - this should bring back more relevant results - or at least results from more context conscious pages.
But for the internet marketer, webmaster or online business owner it has a far more profitable usage, and I'll get to that so hang in there.
A quick, very unscientific and not statistically significant test tells me that Google is using 'OR' operators to rank the 'synonymized' results.
A search for 'software' brings back 123,000,000 results.
A search for '~software' brings back 4,820,000 results.
In the latter you will find 'software', 'freeware', 'shareware' and 'driver' highlighted in the SERP.
Freeware = 6,190,000 results
shareware = 6,490,000 results
driver = 17,900,000 results
A search for 'software OR freeware OR shareware OR driver' brings back 7,480,000 results.
Now there's obviously a big difference between the initially estimated 4,820,000 results and the 7,480,000 results, but even results ranked between 400 and 500 show a nice correlation between the two search result sets.
Now, it's one thing that Google use the 'OR' operator when presenting the search results, but the really interesting thing is HOW it came up with the synonyms in the first place.
Nowhere on the advanced search page or in the FAQ does it say HOW Google decides which words are synonymous with your original keyword.
Google obviously can't use a normal thesaurus because it would be permanently out of date, but with the acquisition of Applied Semantics they got a hold of the massive ASO (Applied Semantics Ontology) which they most probably use for this new feature.
In the ASO, 'tokens' (words) are grouped together in 'terms' which are 'meaningful units' and each term is associated with one or more meanings or concepts.
In the same system, meanings are also related to one or more terms, which is what can be considered synonyms.
So since both the synonym operator and Google AdSense ads are using the ASO it makes sense to use the synonym operator to come up with additional keywords - keywords that can strengthen the theme of your page and make sure that AdSense selects the most appropriate ads for each page.
You can of course use Theme Master to do more extensive theme-research to get your primary targeted keywords the most appropriate context to 'live in', but if you just need a single secondary keyword to strengthen the theme of your page then Google synonyms should do the trick.