A program that cannot be written in (simply-)typed lambda calculus but only in lambda calculus or Turing-complete language
The distinction isn't just about being able to write non-terminating programs. There are significant differences even between normalizing lambda calculi. For example, consider System F. It's normalizing, so all its programs are terminating (so it's not Turing complete), but it has much more expressive power than the simply-typed lambda calculus (it can express any function that can be proved total in second-order Peano arithmetic).

Perhaps the simplest example is Church numerals. In System F Church numerals can be expressed as terms of type $\forall\alpha. (\alpha\to\alpha)\to(\alpha\to\alpha)$. Each natural number $n$ is then represented as
$$\Lambda\alpha.\lambda f:(\alpha\to\alpha).\;\lambda x:\alpha.\; f^n x$$ where $f^n x \equiv \underbrace{f(f(...f}_{n\mbox{-times}} (x)...))$. In the untyped lambda calculus it would be just $\lambda fx.f^n x$.

However, we cannot represent Church numerals and operations on them in the simply typed lambda calculus, because it lacks the necessary polymorphism.

Church numerals can be viewed as one of the simplest recursive data structures. In Haskell we could represent them as


data Nat = Z | S Nat

foldNat :: Nat -&gt; (r -&gt; r) -&gt; (r -&gt; r)
foldNat Z     f z  = z
foldNat (S n) f z  = f (foldNat n f z)


(Observe that foldNat actually converts our representation into the System-F representation.)

System F and the untyped lambda calculus allow us to represent any (inductive) recursive data structures such as list, trees etc. in the similar way as we represented Church Numerals (see System F Structures). This is not possible in the simply typed lambda calculus - it can express only non-recursive data structures with bounded size.

A good source of information is Proofs and Types by Jean-Yves Girard, Yves Lafont and Paul Taylor.