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From what I understand, both can mean activity, but действие also means action.
However, they seem similar enough that I don't know when to use which.
According to this frequency list, действие is ranked #83 among Russian nouns, #302 overall. деятельность is ranked #94 among Russian nouns, #322 overall. Given the prevalence of these 2 words, I believe a deeper understanding of when to use each is necessary.
Деятельность is mostly about working activities, while действие is about actions and functioning activities.
But note that the notion of working activity is rather vague. For example, higher nervous activity = высшая нервная деятельность, that is brain really works, not just functioning etc.
Действие means that the process will be short. Деятельность is the long-running process.
For example:
Действие is a single act while деятельность is a number of different actions with (or without) common purpose.
So, деятельность can consist of many действие.
I believe действие is action, while деятельность is activity. And I can't remember any exceptions.
