All mathematical formalizations of (intuitive) computability are known to be equivalent, in particular they are all equivalent to computability on the universal Turing machine. So technological implementation is irrelevant. The Church–Turing thesis states that this coincides in scope with what is "computable by a human being" unconstrained by limitations of time and resources. This is obviously a philosophical statement, and some disagree, albeit a small minority, but notably early intuitionists like Brouwer and Weyl, Gödel and Lucas. According to them human mind exhibits an ability to creatively "transgress" algorithmic computation, see Does Gödel's argument that minds are more powerful than computers have the inconsistency loophole? More recent intuitionists/constructivists, with the exception of Dummett, generally accept it.
Computable numbers are the real numbers for which there is an algorithm that, given desired precision, returns approximation of the number to that precision in finitely many steps. Those are the real numbers that constructivists and finitists would allow in their versions of real analysis. More broadly, extensions of this line of thought to epistemology are discussed under Do limitations on computability and computational resources have any consequences for epistemology? See also What are the "undefinable numbers" in real analysis and philosophy? for a related class of real numbers.
The main difference is that definable numbers may utilize the full strength of set theory, whereas computable numbers are confined to the resources of arithmetic. Here is an example of uncomputable but definable number. Enumerate all well-formed sentences of arithmetic by Gödel numbers, and define a real number between 0 and 1 as follows: the n-th digit in its decimal expansion is 0 if the n-th sentence is false, 1 if it is true, and 2 if n is not a Gödel number of a sentence. It is definable because the truth predicate of arithmetic is definable (in set theory, not in arithmetic), and we can even easily compute some of its digits, e.g. those that correspond to proven theorems or their negations. But it is uncomputable because there is no algorithm to decide the truth of say the Goldbach or the twin primes conjectures. It is interesting that while constructivists and intuitionists would reject existence of this number early intuitionists did accept existence of unindividuated uncomputable real numbers based on the direct intuition of "becoming" lawless sequences, see Is Aristotle's resolution of Zeno's paradoxes vindicated by motion in the intuitionistic continuum?
