what is this piece for

Yes, the one inside the distributor is connected in a different part of the circuit (coil -) and functions differently; it is there to protect the points from arcing and burning. (And I suppose it could be connected to coil - external to the distributor but I don't recall seeing that on older cars.) It does indeed have to be there; the interference suppresor on the + side of the coil does not.

I used to do marine mechanical/maint in another life and I would diagnose a bad condenser by waiting till the motor started missing and then connect a condenser by hand grounding the case to the coil braket and the pos lead to the to the coil negative.
If it straigtened out the condenser was failing.
Of course this was if there were no other problems I could see for a boat to start running like crap and die 5-10 miles out for no apparent reason.

There were only two times I ever needed to do it that way because the condenser is such a cheap part and should always be replaced along with the points anyway.
Problem was, after paying $1,500 per tow twice because his boat quit 10 miles out the SOB wouldn't believe me that they can do go bad and that it should have been replaced when he did his own tuneup. (A $3,000+ dollar tune up plus a $1.19 condenser now) :D
Cheap bastage had money too.

Anyway, it's the same with car engines that are points fired.
(you can ground the condenser case and touch the lead to any negative part of the wiring between the points and distributor) even with the existing one still in there, and if the one in the distr is bad it will smooth right out as soon as you connect it.

Old School Tricks.