What are some concrete examples of theorems which can be deduced from the Arthur trace formula, which do not follow from the simple trace of Kazdhan and Flicker?
(So I do not mean weaker in the sense that the Arthur trace formula implies the simple trace formula.)
I must have missed this question a month ago, but hopefully you're still interested.
I haven't read that particular paper, but simple trace formulas have restricted test functions, which for one can only deal with representations which are supercuspidal at some place. So any theorem you prove, say about transfer of representations, needs to be restricted to functions which are supercuspidal at some finite place. In general, there may be some other restrictions also, but that depends upon the details of the simple trace formula at hand.
The simple trace formula in Flicker-Kazhdan is exactly what you get when you restrict the full Arthur trace formula to "simple" test functions -- ones where many of the terms on both sides vanish. Importantly, all the complicated terms that are achieved only through regularization vanish, so that the proof of this simple trace formula is also simple.
On the spectral side, only representations with a supercuspidal component appear; on the geometric side, only regular orbits appear. Any result that needs to account for the trivial representation or the orbit {1} won't work with this simple trace formula.
Kottwitz's proof of the Tamagawa number conjecture uses a different simple version of the Arthur trace formula. This one includes the term for the orbit {1}, an essential point since that is the term with the Tamagawa number for the group.
This other simple trace formula can be stated as simply as the one in Flicker-Kazhdan, but is more general as it requires less about the test functions. Its proof requires first constructing the full Arthur trace formula and then verifying that the messy terms vanish. It still won't handle the trivial representation.
This is an addendum to Jason's answer.
Mueller [1] and Lapid-Mueller [2] have exploited Arthur's trace formula to come up with a Weyl law for GL(n) with an excellent error term. (Their method has been adapted by others, Matz, Matz-Templier etc.). Since they are counting all automorphic representations (with a fixed infinitesimal character), the simple trace formula simply cannot detect them.
Much more importantly, Arthur's trace formula has been an indispensible tool in proving important (endoscopic) cases of Langlands' functoriality which hasn't been done (cannot?) by other trace formulas. We now have functoriality so far for classical groups, unitary groups, their inner twists, spin groups.
Nevertheless, Labesse-Mueller proved a weak Weyl law and also Kottwitz could prove the Tamagawa number conjecture (see Jason's answer) using weak test functions in Arthur's trace formula.
[1]: Mueller, Weyl's law for the cuspidal spectrum of $SL_n$
[2]: Lapid-Mueller, Spectral asymptotics for arithmetic quotients of ${\rm SL}(n,{\mathbb R})/\rm{SO}(n)$
[3]: Labesse-Mueller, Weak Weyl's law for congruence subgroups
