Fine needle aspiration Cytology and Aspiration Biopsy would refer to same techniques.
Aspiration biopsy used to be traditionally performed using large bore needles. Using large bore needles resulted in problems such as tumour seeding and complications. Hence the technique was discarded until it was rediscovered in 1950s.
Fine needle aspiration Biopsy (FNAB) (as against fine needle aspiration cytology - FNAC) is sometimes used to refer to Needle Core Biopsy (NCB) where a "cylindrical" bore of tissue is biopsied and the core is processed as a solid biopsy tissue. (Cytology specimens are handled as if they are blood smears!)