The climax of a story seems like a straightforward thing to identify but many readers can be confused about where the climax really occurs. This confusion is due to the vocabulary of the standard definition of "climax," which often says it is "the moment of the highest interest and greatest emotion" and "the turning point of the story." These are perfectly good definitions, if you know what they mean already.
To put it a little differently, a literary climax is the moment or event that is sandwiched between the rising action (and crisis) and the falling action. Bear in mind that the falling action leads to the resolution (the "Whew!" moment) of the story.
Sometimes the climax is defined as the moment with the greatest emotion or greatest tension. One problem with this is that it can cause the climax to be confused with the crisis, which also has great emotion. Looking at a literary climax another way, the climax is most easily recognized because what comes before it gets increasingly difficult, scary, intense, unhappy, lonely, etc. and what comes after it gets decreasingly difficult, scary, intense, unhappy, lonely, etc.
Two other things that can confuse recognition of the climax are anticlimax and resolution. An anticlimax can confuse identification of the climax because it, too, has a fall in action, but the anticlimax always, by definition, falls suddenly into the ridiculous or the comic. The resolution can confuse identification of the climax because it, too, can hold a "most important" part of the story. For example, in one sense, the most important part of the story in The Sphere is when the Sphere from future time escapes from the ocean and explodes into outer space; but the resolution is, by definition, when the problem is solved in one form or another and done away with, like the Sphere is done away with via its escape into outer space.
To recap, the climax follows the rising action and the climax leads into the falling action (not to be confused with an anticlimax...). To be a bit more specific, rising action leads directly to the crisis. The crisis precipitates the climax (think of it as a plateau at the summit of a steep mountain). The crisis comes just before the climax and, in some instances, the crisis may happen at the same time as (simultaneously with) the climax. The climax opens the way for the falling action. The falling action ends in the resolution where the problem is done away with, which can sometimes be a very important event or moment. The best way to look for the climax is in the right location, not by emotion, intensity or importance.
Having said that, the climax of Isaac Asimov's story "The Robot that Won the War" is when Swift tells Henderson that he didn't use Multivac's data output (which seemed to him to be improbable and sometimes impossible) for making decisions on war strategy but that he used the toss of a coin instead. This climax is a surprise ending that doesn't accommodate classic forms of falling action and resolution.