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.
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.
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.