Dismiss, discharge, cashier, drop, sack, fire, bounce are comparable when they mean to let go from one’s employ or service. Dismiss basically denotes a giving permission to go.
When used in respect to employment it carries apart from the context no suggestion of the reason for the act and is, therefore, often preferred as the softer or as the more comprehensive term.
Discharge is usually a harsher term, implying dismissal for cause and little or no likelihood of being called back.
Only in military and court use does it, when unqualified, carry no implication of dissatisfaction on the part of the employer.
Cashier implies a summary or ignominious discharge from a position of trust or from a position that is high in the scale.
Drop, sack, fire, and bounce are all rather informal.
Drop is a common and colorless synonym of dismiss.
Sack stresses a being discarded or thrown out of employ while fire stresses a dismissal as sudden and peremptory as the action of firing a gun and bounce, a kicking out.