It is fair to say that the concepts are "equivalent" in the modern mathematical practice. However, they have different histories and different overtones of meaning. The notion of natural or counting numbers goes back (officially) to Pythagoreans and unofficially to prehistoric times, Ishango bone, a counting artifact, is 20,000 years old. We are dealing with a loose concept imported to mathematics from natural language after centuries of practical and then more abstract use. While in one sense cardinals generalize natural numbers, in another sense they are a much more narrow, technical, and context bound notion.
The notion of "cardinal number" or "cardinality" (even finite) is much more recent, it is one way to make the everyday concept more precise for mathematical use, the one favored by the current formalization of mathematics. The notion was introduced by grammarians in 1590-s to distinguish between counting numerals (one, two, three) and order numerals (first, second, third), so-called ordinals. But their modern mathematical life begins at the end of 19th century in Cantor's set theory, which imported the distinction and made it even more technical. Cantor replaced the elementary concept of a number with a derived and abstract one based on sets (for what it is worth, the informal counting conception is closer to his ordinals than to his cardinals, the notions are equivalent in scope in the finite case but diverge dramatically for infinities). Two sets are equipollent if they can be put into a bijective correspondence (this is now called Hume's principle), and the cardinal number of a set is something like the equivalence class of all sets equipollent to it, or so Cantor wanted to say. This turned out to have technical problems, and now more "concrete" definitions are preferred, ones that identify it with one of very specially generated sets like ∅,{∅},{{∅}},... or ∅,{∅},{∅,{∅}},... , see Set-theoretic definition of natural numbers.
This multiple realizability is already a problem for philosophers, we would like to think that there is a unique concept of "two", expressed as 2 or {{∅}} or {∅,{∅}}, but set theory can only provide us with an arbitrarily chosen token for it. This leads to popularity of mathematical structuralism, the philosophical position that it is not what it is that makes 2 a 2 but only its place in a structure, in this case the structure of the counting number series described by the Peano axioms. Peano arithmetic already shows that natural numbers, unlike cardinals, can be lifted off of set theory since it does not depend on it. Neither does category theory, which has its own versions of natural numbers. Even within set theory the notion of "finite cardinal" can come apart from that of a natural number if we play around with the axioms. For instance, Dedekind defined "finite cardinals" as equipollents of those sets that can not be put into bijective correspondence with their proper subsets. Well, Russell and Whitehead showed in 1912 that Dedekind finite cardinals can be infinite (on the usual conception) in set theory without the axiom of choice, so-called ZF, or other alternative set theories. Intuitively, this is because those models of set theory do not have enough bijections to "detect" infinity Dedekind's way.
Cantor's set theory, and even Hume's principle it is based on, were historically controversial and not without alternatives, see Measuring the size of infinite collections of natural numbers: Was Cantor's theory of infinite number inevitable? by Mancosu. If mathematicians decide one day that they prefer categorical or some other formalization of mathematics to the set-theoretic one the notion of cardinal number may go away, but we can be sure that the natural numbers will still be around.
