Refers specifically to the front part of an airplane or similar aircraft.
機首が滑走路に向けられた。
The nose of the aircraft was turned toward the runway.
The pilot raised the nose and climbed.
パイロットは機首を上げて離陸した。
The pilot raised the nose and took off.
The plane turned, pointing its nose to the east.
機体 refers to the whole airframe of an aircraft, whereas 機首 refers only to its front nose section.
Compound of 機 (aircraft) and 首 (neck/head), literally 'aircraft head'.