Graduate Clarification of Mihăilescu's Theorem (Catalan's Conjecture)

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Mihăilescu's theorem confirms that Catalan's conjecture holds true, with the only natural number solution to the equation x^a - y^b = 1 being x=3, a=2, y=2, and b=3. The discussion raises questions about whether Mihăilescu's theorem implies that no other pairs of powers can equal 1 under different conditions. It seeks clarification on whether specific restrictions apply to the integers involved, such as requiring x or y to be prime or if only the conditions x, y > 0 and a, b > 1 are necessary. The conversation references Wikipedia for a more detailed explanation of these restrictions. Understanding these nuances is crucial for comprehending the broader implications of Mihăilescu's theorem.
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I understand Catalan's conjecture was proven by Preda V. Mihăilescu in 2002. However, I am not sure if it is proved for only certain conditions.
Mihăilescu's theorem proves that Catalan's conjecture is true. That is for x^a - y^b = 1, the only possible solution in naturual numbers for this equation is x=3, a=2, y=2, b=3. What is not clear to me is this. Does Mihăilescu's theorem prove that the difference between any other two powers (not the Catalan expression) will never be equal to 1 but only within certain restrictions? Another words, are there conditions that restrict x or y have to be both prime integers or just one of them must be a prime integer or does a or b have to be both prime integers or just one of them must be a prime integer for Mihăilescu's theorem to be true? Or is the only condition necessary is that x,y >0 and a.b >1?
 
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