In most of mathematics (like number theory or analysis), it's the countable infinite objects and their properties that are being studied, like natural numbers or infinite sequences. The finite sets themselves are considered somewhat trivial (natural) objects, and uncountable sets are mostly a universe where things happen. As a consequence, for mathematicians, all natural numbers are the same, no matter how large.
In computer science, I would argue, the view is quite different. The interest of computer science is in medium-sized but finite sets (roughly between 2^6 and 2^(2^6)). The small sets of size less than 2^6 are trivial (you can brute force them without a computer), while with sets that are larger than 2^64 we almost don't need to compute with (they are like limits what can a DB store), and numbers above 2^(2^64) are pretty meaningless. So you can see, different natural numbers matter differently, some are important and some are less, they are not treated as identical objects.
Since computer science is kinda part of mathematics, you can see the philosophical focus is different.