In both cases, the employee leaves the organisation. Discharge- is voluntary separation, Dismissal- is removal by the organisation as punishment, Layoff / downsize - is the removal on terms of economic/financial situations with or without compensation.
Here are some examples:
His resignation letter was accepted and he was relieved from the services.
As his forgery of company documents was proved, he was immediately dismissed and discharged from the services.
As the subordinate misbehaved with the manager, he was suspended for a week.
They were laid off the services from Dec 31.
'Laughing' and 'Smiling' have different situational terms, though they exhibit ' a curve on lips'.
Hope I have tried to explain you.
Rgds
Chandru