Is it the same to say the following?
He was discharged from the police force for bad conduct.
He was dismissed from the police force for bad conduct.
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Good question. And J.R.'s comment is very useful (+1), they are not always interchangeable. On the other hand I do believe that both the words here, in this context are okay to interchange. That's because I referred dictionaries and found that discharge and dismiss both can mean terminate the employment of a person.
However, if I apply my logic, a subtle difference comes in my mind. When you discharge someone, you emphasize more on their duty or work/task they do. Think this way - if a commissioner comes, he takes charge of the ex-commissioner's duties. This way, if he's discharge, we emphasize more on his duties (of course, he won't remain on his seat).
Now, dismissal. Dismissal I think emphasizes more on the position than his duties.
[Can we think of (I'm asking this) this - Alex was discharged (emphasizing on his performance, duty, job, capability) from that crucial project as he wasn't doing good on the field and allotted with clerical work now in the office. On the other hand, Alex was dismissed (emphasizing on the position/designation) from the company as he was good at nothing.]
Again, mind it, it's a subtle difference and after dismissal or discharge, the person is terminated from the employment.
In this context, the two have very similar if not identical meanings.
From the definitions of both words, you can see that to 'discharge' is an official demand for someone to leave, while a 'dismissal' can mean both 'letting someone go' and 'sending them away', or even 'allowing someone to leave'.
The reason they are identical is simply because 'discharging' someone from a police force or armed forces unit etc, is defined as a dismissal. The two words have become interchangeable here.
The two words are close in meaning. When talking about someone leaving a job, being dismissed has a negative connotation. (Let's ignore other usages of the two words. Your comment suggests that you already know them.) We can boil the difference down to this:
discharge - allow someone to leave their job
dismiss - make someone to leave their job
Essentially, it's about "allow" vs. "make". You can find lengthier definitions in dictionaries for learners. For example, Macmillan Dictionary defines
discharge (sense 1) as "to be officially allowed or forced to leave an institution such as a hospital, a prison, or the army"
(so discharge can mean either "allow to leave" or "force to leave"), and
dismiss (sense 2) as "to force someone to leave their job".