This is an interesting find, try searching on the keyword web and you should see 435 million results found. Now add the parameter &client=firefox to the Web search and you will find 185 million results. I am sure you get slightly different results then what I reported, but the fact of the matter is, by adding &client=firefox to the search URL, you will get less results then by not having it. It works the same by putting anything in the parameter, its not just a firefox deal.
The best theory, so far, IMO, as to why this happens is:
The obvious theory is that "client=" is a flag Google uses for something other than a browser type.
At a quick look, adding "client=1" seems to return approximately the same number of results as using Safesearch.
Guess: perhaps "client=whatever" refers to queries coming from third parties using Google as a back-end? Google is filtering the results to avoid offending their user base.
Forum discussion at WebmasterWorld.