What does the continuity of $f'$ tell us about $f$? Suppose $f$ is differentiable on $\mathbb{R}$ and its derivative $f'$ is continuous on the interval $[a,b]$. What constraints on $f$ would such condition give us? 
 A: Here is an answer but I do not want it upvoted since it is not in the spirit of the question.  What we really want is some property expressed more directly in terms of the values of $F$ that is both necessary and sufficient in order that $F'$ exists and is continuous.  I offer this a "close but no cigar."

Theorem. A necessary and sufficient condition that $F:[a,b]\to \mathbb R$ is continuously differentiable on $[a,b]$ is that $F$ has a
  strong derivative at every point of that interval.

The strong derivative was introduced by Peano in 1892 (who called it a strict derivative and recommended that it should be used in elementary courses instead of the ordinary derivative).  Unfortunately the "strong" terminology sticks, although "strict" would have been better.  It is defined as
$$D^\sharp F(x) = \lim_{(u,v)\to (x,x)} \frac{F(v)-F(u)}{v-u}$$
and sometimes called an "unstraddled" derivative since the limit here is not computed only for intervals $[u,v]$ that contain or straddle  $x$, but for all "close" to $x$.
It has appeared before in a StackExchange question where Dave Renfro supplies a (probably) complete bibliography on the topic.
An equivalent answer, but sufficiently disguised that it may appear profound, is to express this in terms of the Dini derivatives.

Theorem. A necessary and sufficient condition that a continuous
  function $F:[a,b]\to \mathbb R$ is continuously differentiable on
  $[a,b]$ is that at every point of that interval   one at least of the
  four Dini derivatives is continuous.

That is also not particularly intrinsic.
I do have an intrinsic answer in mind but it is not going to make the OP happy so maybe this weak answer suffices.
