Although thrillers and crime fiction may be classified as mystery, they each have their own specific requirements. Thrillers--sometimes called suspense--utilize tension and excitement as their main elements. Often they involve time-deadlines, spies, heists, conspiracies, cover-ups, fights and/or chase scenes. The most important point about thrillers is the characters act to stop a crime before it happens. Notice, the protagonist(s) and the reader know what this impending crime is and it is very large in scope. Thus, climax of a thriller is when the protagonist(s) defeats the antagonist(s) to stop the impending crime. Thrillers often utilize multiple 3rd person pov characters and can include the pov of the villain/antagonist. Crime fiction is typically centered on criminal enterprises and is told from the point of view of the perpetrators.
In mystery fiction proper there must be a serious crime--usually one or more murders--in chapter one and the characters try to discover 'whodunit'. Typically, a vital piece of information is kept mysterious until the climax when the investigator solves the crime. Of course, there must also be an investigator who can be a professional detective, a hard-boiled private eye, a forensic anthropologist, a klutzy bounty hunter, a psychic waitress or whatever. :) In recent years, so-called cozy mysteries have abounded, filled with recipes, talking cats, knitting, white-water rafting or you-name-it; these involve amatuer sleuths who have to solve the crimes because of circumstances, they find a body in their garden, for example. Often, mysteries have one first-person pov character (the sleuth). According to the-powers-that-be, the market for straight-up cozies is supposed to be waning.
For historical reasons, it's important for mystery writers to give the reader the chance to solve the mystery. Make sure the reader has the chance, by being given all the necessary info, to beat the sleuth to the finish line. If you don't, readers will be upset! :(
Notice, too, mysteries are very moral. In the end, the bad guy is caught and is, or will be, punished, and the bodies are buried with respect.
A third component of mystery fiction is social commentary. With a signicant crime and a well-characterized investigator, however, a socially-relevant examination of humanity should easily follow.
The Mystery Writers of America is the premier organization for mystery and crime writers, professionals allied to the crime writing field, aspiring crime writers, and folks who just love to read crime fiction. Check it out!