Both are connected at one point , however the PE can (and will) carry more noise due to currents (up to 30mA or even 300mA depending one the installed RCD, in some cases even higher ) from line filters in switched power converters , C&L coupling to the line power etc..
While the PE is build (and tested! ) for low resistance the impedance (here inductivity) can be quite high, so RF content is not well grounded.
So a instrument earth as an separated path is build up, but not with the intend to carry lot of current, just to establish a clean ground potential.
To Q2: A capacitor to ground will be low impedance for high frequency (Z=1/(j*2*pi*f*C)) but not for DC . On the other hand an inductivity is a good conductor for DC , but not for AC (Z=j*2*pi*f*L).
Depending on the task you have, you can decide what part you tide to ground.
Hendrik (almost) said it all. A PE terminal should not be used as return path or GND terminal for signals. Its main purpose is to supply a connection between the device housing and the earth line of the mains power supply.
Sometimes it is necessary to make a connection between the signal return path and the GND or earth terminal, e.g. in case there are EMI problems. In this case sometimes it helps to connect the signal return path or signal GND with a (rather small) capacitor to the earth terminal.