Cognitive Computing vs Artificial Intelligence?
A brief definition would be:


  Cognitive computing is the simulation of human thought processes in a computerized model.


more detailed explanation:


  Cognitive computing involves self-learning systems that use data mining, pattern recognition and natural language processing to mimic the way the human brain works. The goal of cognitive computing is to create automated IT systems that are capable of solving problems without requiring human assistance.


Cognitive computing is used in A.I. applications. Therefore I think it can be concluded that Cognitive Computing is a subset of Artificial Intelligence.

Cognitive Computing systems may include the following ostensible characteristics:


Natural Language Processing
Machine Learning
Algorithms that learn and adapt
Vision-based sensing and image recognition
Spatial and contextual awareness
Reasoning and decision automation
Sophisticated pattern recognition
Neural Networks
Semantic Understanding
Noise Filtering
Common Sense
Robotic Control
Emotional Intelligence


So basically, you can say that a cognitive computing system might be trained by neural networks.

Sources:


http://www.cognitivecomputingforum.com/
http://whatis.techtarget.com/definition/cognitive-computing