A Process engineer is usually an engineer that is given the responsibility to follow one or more process units. They typically plot operating data and compare it with the planned operation - typically from an LP based plan. They usually are responsible to let management and planning personnel know the status of the unit (equipment and performance status at daily meetings).
Operation engineers are usually more involve with the status of the equipment often focusing on mechanical integrity rotating equipment (pumps and compressors) any other equipment that requires significant maintenance that could cause a slowdown.
The two roles often overlap based on the unique needs of each plant – equipment that limits profitability or has significant safety issue will get more attention from all engineers and operators than equipment that operates smoothly and doesn’t usually impact profirability or reliabilty of the entire plant.