I know that "we can construct" is a stronger claim than "there exists" because sometimes existence is known but explicit examples/constructions are not known.
I am looking for a deeper understanding of this. Is this difference always related in some way to the axiom of choice? Or maybe it is always related to the concept of computability?
My thoughts about the connection with the axiom of choice
I have a superficial understanding of the axiom of choice: I know that it is necessary in some situations about picking elements of set(s), i.e., it is not always true that we can simply "pick" an element (usually when the set in question is very big or it lacks a clear method of choosing). Intuitively this might be related to my question because it seems that "there exists" is weaker statement in the sense that "it exists" but "we don't know how to pick it".
My thoughts about the connection with computability
There are several examples of things that exist but are uncomputable, and therefore there is not an explicit way to "obtain", "pick" or "construct" them, such as chaitin's constant. This is why I think this is related, but I'm not sure, because perhaps it is more related to "definability" than computability (several uncomputable numbers are at least definable).
For all situations in which "there exists" applies but "we can construct" doesn't:
Is the axiom of choice (in any of its forms) necessarily being used?
Is the thing that exists but can't be constructed always an example of an incomputable thing?