Michael Morgan, 1/5/2019
It is easy to confuse the two, since both fields seek to understand the cognitive (as well as affective) processes in the human brain.
1. Cognitive Neuroscience has a strong empirical focus, with an interest in and usage of specialized brain scanning (CT Scans), neural pathway mapping (PET Scans) and detection of abnormalities (MRI Scans). In addition cognitive neuroscientists use other physiological measurements (e.g., heart rate, eye movement, stress levels) to help understand specific brain processes.
2. Computational Neuroscience is described fairly well in the website for the Organization for Computational Neurosciences:
Computational neuroscience (CNS) is an interdisciplinary field for development, simulation, and analysis of multi-scale models and theories of neural function from the level of molecules, through cells and networks, up to cognition and behavior.
We in CNS work closely with experimental data when available at various scales -- CNS models these data to allow them to be understood clearly and make predictions for new experiments.
Identification of scale interactions and dynamics in neural structures provides a framework for understanding the principles that govern how neural systems work, and how things can go wrong in brain disease. (This points to a specific focus on brain diseases, which most often translate into behavioral disorders. Thus, Computational Neuroscience often looks for ways to track and treat pathologies emanating from the human brain.)
CNS links the diverse fields of cell and molecular biology, neuroscience, cognitive science, and psychology with electrical engineering, computer science, mathematics, and physics. (This is the most important differentiating factor. Modeling mental and behavioral phenomena is not restricted to strictly empirical studies of the brain, but instead can and should leverage the tools available in machine learning, mathematical simulation and optimization, and areas concerned with equilibrium and off-equilibrium brain processes as these can be measured at various levels of aggregation, from individual to epidemiological, using the mathematics of neural networks and other latent state, graph-based models of trends and transformations over time.)