What are the key differences between warm and cold core cyclones? What are the key differences between warm and cold core cyclones?  What is an example of each?
Key differences are in structure, location, environments they thrive in.  To compare/contrast these storms I'll use a mid-latitude cyclone for the cold core cyclone and a hurricane for the warm core cyclone.

Structure
The hurricane is nearly symmetric and devoid of fronts.  The mid-latitude cyclone is asymmetric (comma shaped with a long tail) and has fronts associated with it.

Location and Movement
Hurricanes form in the tropics (but not around the Equator) and initially move from east to west with a discrete well-defined low pressure center.  Hurricanes move with the wind.  Mid-latitude cyclones occur in the mid-latitudes and move west to east with centers of low pressure that may dissipate and re-form further downstream.  Mid-latitude storms move as temperature advection drives pressure falls ahead of the storm.

Environment
Hurricanes need warm ocean surface and weak vertical wind shear throughout the troposphere to develop.  Mid-latitude storms are at home with shear and require horizontal temperature gradients to develop.


Hurricanes can transition into cold-core cyclones once they exit the tropics.  It is possible, but not common, for a mid-latitude cyclone to transition into a warm-core tropical cyclone.