Why does reversing the order of these two transfer functions give me different outputs?
If the input is a unit step, then the output of the first block in system 1 is not zero, but it is a Dirac delta impulse $\delta(t)$. Intuitively, the derivative is infinite at $t=0$ because of the step going from zero to one. Integrating the Dirac delta impulse will give you a step at the output of the first system.

The output of the first block in system 2 is a ramp, which will be transformed into a step by the differentiator. So both systems have the same output.

Note that in general you can only interchange the order of LTI systems without changing the output if all systems are stable. For unstable systems, as in your example, the order can make a difference, because the corresponding integrals might not converge, and consequently you can't change the order of integration.