First and foremost I have the updated bode plot with the mode moved to 10 Hz. See Attachment 1. Note that the comparison measurement is a % difference whereas in the previous bode plot it was just the difference. I also wrapped the phase so that jumps from -180 to 180 are moved down. This eliminates massive jumps in the % difference. 

Next, I have two comparison plots: 32 bit and 64bit. As mentioned above I moved the mode to 10 Hz and just excited both systems at 3.4283Hz with an amplitude of 1. As we can see on the plot the two models are practically the same when using 64bits. With the 32bit system, we can see that the noise in the IIR filter is much greater than in the State-space model at frequencies greater than our mode.

Note about windowing and averaging: I used a Hanning window with averaging over 4 neighbor points. I came to this number after looking at the results with less averaging and more averaging. In the code, this can be seen as nperseg=num_samples/4 which is then fed into signal.welch