Apology for the vague question, but I hope I can make it clearer by considering some examples. Also, I added algebraic geometry to the tags because I'll borrow my examples from there, but feel free to change the tags.
From a formal point of view, an equivalence between two categories tells us that the two categories are 'essentially the same'. Nevertheless it may still happen that two equivalent categories come from different fields of mathematics, both with their own sets of definitions or theorems, which do not seem to translate to each other quite easily. Here are three examples which exhibit the just-mentioned phenomenon in different degrees.
The category of prevarieties over an algebraically closed field $k$ is equivalent to the category of integral schemes of finite type over $k$. (The definitions are as in Mumford's Red Book.) In both categories, objects are represented by topological spaces with a sheaf. These spaces are constructed very similarly (the former from maximal ideals, and the latter from prime ideals) and look very much alike (the latter being like the former but with some hidden points, some reducedness). Many definitions, constructions, and proofs in the former category just carry over to the latter.
The category of smooth projective curves over $\mathbb{C}$ is equivalent to the category of compact Riemann surfaces. There are strong similarities between the two categories. Both have meromorphic functions and divisors, both have differential forms, both have a Riemann--Roch theorem, and so on. It seems to me (but I may well be wrong here) that these similarities are more of a heuristic observation than something formal. For instance you can't apply the analytification functor to transform Riemann--Roch for curves into Riemann--Roch for Riemann surfaces, let alone transform a proof of one into a proof of the other.
For many suitable ringed spaces $(X,\mathcal{O}_X)$, there's an equivalence between the category of finite-rank locally free sheaves on $X$, and the category of finitely generated projective $\mathcal{O}_X(X)$-modules, including for affine schemes and smooth manifolds. The objects representing both categories are very different in nature, and it seems to me that concepts from one category don't have a nice translation to the other category. (Stupid examples: How do I say that a smooth vector bundle $E$ on $M$ has vanishing curvature in terms of $C^\infty(E)$? How does saying a projective $A$-module is the natural module over a matrix ring translate to coherent sheaves?)
There is an equivalence between the category of commutative $C^*$-algebras and the opposite category of compact Hausdorff topological spaces. Yet the objects representing the two categories are entirely different. There are concepts in one category that do not translate well to the other. (Stupid examples: To which $C^*$-algebras do manifolds correspond? What is the Spectral Theorem in the context of compact Hausdorff topological spaces?)
It is often said that the distinction between classical and modern algebraic geometry is artificial, and the equivalence between classical varieties and finite-type integral schemes should back up this statement. But no-one would say that the distinction between functional analysis and topology is artificial because of Gelfand--Naimark. So how much should I still think of the equivalence between varieties and integral schemes? Exactly how happy should I be when I prove that two seemingly different categories are equivalent? In short:
What does an equivalence of categories really tell us?
