The Fertile Crescent is located in parts of Israel/Palestine, Iraq, Jordan, Lebanon, southeastern fringe of Turkey and, the western fringes of Iran. Although, it is true that some authors do include Cyprus and parts of Egypt as well.
Mesopotamia is a historical region. It is situated in Western Asia, within the Tigris–Euphrates river system. Nowadays, you can consider that it corresponds to most of Iraq, Kuwait, the east of Syria, Southeast of Turkey, and parts of Northern Saudi Arabia.