Public Affairs - TF force can calm population of the system with more effectively.
Operations (Operations Officer only) - will help Commander with crew training (and no more).
Intelligence - increases chances of getting intel info about enemy at the time of battle.
Actually, commander/staff bonuses in this version are quite useless, with the only exception of Crew Training and (for the staff officer) Operations.
And, worse of that, Fighter Operations bonus is bugged, so it works negatively (more bonus - more time needed to rearm in hangar). Will be fixed in the next version (C# Aurora).