What can be done wrong changing the chain in a single speed?
The photos show that the contact surfaces of the cogs have been worn to match the old chain. The new chain does not match them perfectly, which is the reason to the noise. 

For some reason singlespeed seems to be much more sensitive than derailleur cogs. The good news is that if you can tolerate the noise and vibration for a while, the new chain and old cogs do wear to match each other. If you do not want to do this, replacing the freewheel should help.