Is there a difference between filtering a signal before or after differentiating it?
So, first of all:

A differentiator is really just a high-pass filter. Digitally (and this is dsp.SE, so I presume this is the case), that typically means a differentiator is really just FIR with taps $[1;-1]$. A butterworth filter being a low-pass IIR, the combination of both does sound like you might want to build a bandpass, or are doing something that doesn't make very much sense; see the comparison of a very relaxed second order butterworth and a differentiator magnitude responses below:



now, mathematically, that's identical: 

In time domain, you convolve ($*$) the signal $x[n]$ with the filter impulse response:

$$\begin{array}\\
y[n] &amp;= x[n] * h_{diff} * h_{butter}\\
&amp;\text{or, in freq. domain, thanks to the convolution theorem}\\
Y[z] &amp;= X[z] H_{diff} H_{butter}\\
&amp;\text{multiplication is commutative}\\
&amp;= X[z] H_{butter} H_{diff} \\
&amp;\text{but it's also associative}\\
&amp;= X[z](H_{butter}H_{diff})\\
&amp;= X[z] \mathcal Z\{h_{butter}*h_{diff}\}
\end{array}$$
which implies you can even without problem combine the differentiator and the low pass filter. To little surprise, you get the discrete derivative of the butterworth filter.