This question is a little bit tricky because it's important to remember that Uruk and Uruk-Hai refer to different things, at least in the dialect of the Isenguard Uruk-Hai. Tolkien was a linguist, and he loved adding this kind of cultural quirk. According to legend, he created Middle Earth, just as a place to simulate these kinds of cultural drifts, changes and misunderstandings.
Isenguard Uruk-Hai are strongly implied (in the book trilogy) to be an abominable hybrid of the races of Orcs and Men, designed by Saruman to have all the strengths of both and none of the weaknesses.
Meanwhile Orc, Uruk, Hobgoblin, Goblin and Snaga (lit. "slave" or "maggot") are all words that refer to members of the Orc race, which was created by the First Dark Lord, Morgoth, in the Pit of Angband.
One character might use one word in particular to refer to any Orc, because of that character's cultural background. Another character might use each of those words selectively, depending on the size, heritage and social status of the orc in question.
Some purebred Orc tribes even go so far as to claim the title of Uruk-Hai for themselves, adding another term for pure Orcs, and further complicating the task of distinguishing between Orcs and Hybrids. In universe, it appears that Saruman actually appropriated the term from them, when indoctrinating his hybrid shock troops, but it still makes things extra confusing.