In simple terms they are all pointers to same the functionality and used interchangeably. But if you are looking for some finer details then in 800xA:
Asset: A valuable piece of entity in the plant. This could be a hardware or software. E.g. Pump, motor, field instruments, Computers, Network Switches or simply a windows program.
Asset Optimization: It the umbrella name for the overall product or functionality just like PLC Connect or Information Management.
Asset Management: Managing the plant assets using Asset Optimization products.
Asset Monitoring: Monitoring the condition or health of an asset using Asset Monitors.
Asset Monitors: Actual logic that monitors the condition of an asset, does the calculation and provides detailed information. For example "High Limit Asset Monitor" or a dedicated Asset monitor for a field instrument.