Some context is required here - what sentence would you like to use / translate?

General means applies to many situations. Common means it is readily available or present.

I am not a native speaker, but per my understanding, genera is opposite to specific. When you talk about some topics without giving realistic examples, you are speaking 'in a general way' (or generally speaking, or your speaking is overly generalized). Something common is the thing you normally see or hear, which is ubiquitous Just for your reference

общий и обычный