The simplest answer is given by the fact that typed lambda calculi correspond to logics (simply typed lambda calculus -> predicate logic; system f -> second-order logic) and consistent logics cannot prove their own consistency.
So let's say that you have natural numbers (or a Church encoding of natural numbers) in your typed lambda calculus. It's possible to do a Gödel numbering that assigns every term in System F to a unique natural number. Then, there is a function $f$ that takes any natural number (that corresponds to a well-typed term in System F) to another natural number (that corresponds to the normal form of that well-typed System F term) and does something else for any natural number that doesn't correspond to a well-typed term in System F (say, it returns zero). The function $f$ is computable, so it can be computed by the untyped lambda calculus but not the typed lambda calculus (because the latter would amount to a proof of the consistency of second-order logic in second-order logic, which would imply that second-order logic is inconsistent).
Caveat 1: If second-order logic is inconsistent, it might be possible to write $f$ in System F... and/or it might not be possible to write $f$ in the untyped lambda calculus - you could write something, but it might not always terminate, which is a criteria for "computable."
Caveat 2: Sometimes by "simply typed lambda calculus" people mean "simply typed lambda calculus with a fixed-point operator or recursive functions." This would be more-or-less PCF, which can compute any computable function, just like the untyped lambda calculus.