The main difference between pulmonary and systemic circulation is the presence or absence of oxygen in the blood. Pulmonary circulation is the movement of deoxigenated blood (no oxygen) from the heart to the lungs through the pulmonary artery. When the blood reaches the lungs, oxygen is added and it is sent back to the heart. Then, the heart sends this oxygen rich blood out to the rest of the body. Systemic circulation is the part of the cardiovascular system that carries the oxygen-rich blood to other parts of the body, and returns deoxigenated blood back to the heart.
One mnemonic device you can use to help you remember the difference is that pulmonary sounds like "pull." The purpose of pulmonary circulation is to "pull" oxygen from the lungs. Systemic of course sounds like "system." In order for all of the systems of the body to work, they need oxygen. Systemic circulation supplies oxygen to keep the system running.