re: differential equation solutions, you can compute if they are within epsilon of each other for any epsilon, which I feel is “morally the same” as knowing if they are equal.
It’s true that the concepts are not identical. I feel computability is like the “limit” of the “explicit” concept, as a community of mathematicians comes to accept more and more ways of formally specifying a number. The correspondence is still not perfect, as different families of explicit formulae will have structure(e.g. algebraic structure) that general Turing machines will not.