What are the key differences between warm and cold core cyclones?
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.