PLC is a Programmable Logic Controller, used in industry to control motors and stuff, for example a production line with motors to drive belts, various arms and levers moving etc.
Data logger captures data and stores it for use later on. Say if you want to monitor the temperature in 4 different rooms for 2 weeks at 1 minute intervals, you set up your data logger with temp sensors, set it going and then read back the data after 2 weeks. Then you can check and analyze the data etc. Some loggers have alarms and other stuff as well.