The difference is simple: the material implication has nothing whatsoever to do with a conditional.
It should also be said that we can always use a conditional to express a logical implication, including the logical implications that mathematicians prove that there are from axioms to the theorems that follow from these axioms, and that therefore the material implication has also nothing to do with logical implications in a mathematical context.
The expression "If x = 2 then x² = 4" is a conditional. However, it is also the straightforward interpretation of the logical implication x = 2 → x² = 4. This implication is not formally true, since it depends on a whole set of definitions which are not formally expressed in this expression, but we all understand what those are and we can think of them broadly as the axioms of arithmetic.
It is also apparent in mathematical papers that mathematicians never use the material implication when they prove a theorem. Nearly all mathematical proofs rely on the logical sense of the mathematician, not on any formal calculus based on the material implication.
It should be noted that mathematicians are able to understand each other's proofs simply by reading them, like they have always done since Euclid, essentially in the same way that we can all understand Aristotle's syllogisms, that is, intuitively. Thus, no mathematician would verify a proof by using the definition of the material implication.
Thus, the fact that mathematics usually works well, including in applications in engineering and science, is no evidence that the material implication has any value.