It is easy to confuse two words when they have the same root. This is the case with the adjectives (words that modify nouns or persons, places, and things) incredulous and incredible. As incredible as it might seem and as incredulous as you, gentle reader, may be, these two words are not synonyms, i.e., they do not mean exactly the same thing. Therefore they may not be used interchangeably. Incredulous means “unbelieving.” It describes the subjective state of a person: “He was incredulous at some of the claims being made for that new toy.” We may substitute unbelieving for incredulous: “She was unbelieving when confronted with the manufacturer’s claims about the new medication.” The root word of incredulous is the Latin incredulus. According to the Oxford Dictionary of the English Language (OED), incredulus was used in the 16th and 17th centuries to describe those who were “skeptical of religious belief.” Today, however, the word is used more generically to describe skepticism of any kind. If you intend to communicate skepticism then, of the two, incredulous is the correct adjective.
The second adjective, incredible, would seem to be so similar to incredulous as to be interchangeable but on reflection we see that it is not. Like incredulous, its roots are also in Latin, incredibilis, which, in Latin, means unbelievable” or unbelieving. In both cases we should avoid the etymological fallacy, i.e., determining the sense or meaning of a word by its etymology, e.g., whenever a preacher tells you that dynamis in Greek is related to dynamite (which is true) and then proceeds to use dynamite to define dynamis. So, in this case, incredible describes a person, place, or thing that is beyond belief. It has been used that way in English since the 15th and 16th centuries. Strictly used it describes something that is literally impossible to believe. “Her thesis was incredible” would mean that the argument proposed was literally beyond belief because the evidence did not sustain it.
Most often, however, the word is used to express hyperbole: “That 75 foot putt was incredible!” If we all saw the same putt, then, of course, it was not literally incredible. It was amazing, it was stupefying, it was remarkable, it was thrilling but it was not literally incredible. In this sense, as the OED notes, the author or speaker means to say that, prior to the event one would have thought it incredible.
My English friends use incredible in the strict sense of being beyond belief. American usage has tended toward the second, hyperbolic sense for decades.
When you want to communicate that you (or others) do not believe something, use incredulous. When you want to say that something or someone is either beyond belief or perhaps so unusual as to be difficult to believe or hitherto impossible to believe, use incredible.