May I answer your query in relation to the British Army in simple terms:
Promotion - move to the next paid rank - i.e Corporal to Sergeant; Lieutenant to Captain
Appointment - is usually a move to a specific job or position in the battalion, such as Master Cook, Master Tailor but paid as a Sergeant. Examples: Frank Bourne was promoted Sergeant (15 Apr 1878) and appointed CSgt B Coy 2/24th (27 Apr 1878). It is quite likely that he was only paid as Sergeant at the time of the defence of Rorke's Drift. Teignmouth Melvill was promoted Lieutenant (2 Dec 1868) and appointed Adjutant 1/24th on 7 Mar 1873. Lance Corporal and Lance Sergeant are appointments as they are still paid as Private or Corporals.