Two people recently asked me to explain hashtags on Twitter (mostly related to search queries). To best understand that it's a matter of context, take a look at the following 2 examples, each with and without a hashtag.
Search "#Random": people tweeting about things that are random; random is the subject, or tag, the defines their tweet.
Search "Random": the word random so happens to be a word in their tweet and therefore displays in the results page.
In the above case, the # is important cause the context of the word is different in both cases.
Search "#fml": people tweeting about how shitty life is (fml = fuck my life; originated from www.fmylife.com); fml is the subject, or tag, that defines their tweet.
Search "fml": pretty much pulls up the same types of tweet cause fml only has one meaning so even if it's used without the #, it still pulls up similar results.
So, at the end of the day, the hashtag will likely give more accurate search results since people are tagging their tweet with a specific subject, vs just using the word in there.
I think the 2 images speak for themselves (click to enlarge).