The “Metro” area generally refers to the main city and any other high density cities right beside it, probably linked together by high volume public transit with massive numbers of people commuting between each other every day.

The “Greater” area generally refers to a much wider area including many of the much smaller towns that even people in the same country won’t recognize, rural farmland, etc. Their economies and business are probably mostly dependent on the main city so they may be considered linked, even though there aren’t as many people moving back and forth between them daily.