There isn't much difference. Both have multiple meanings, and they are nearly the same meanings. It's just that the likelihood of using one of these instead of another for a given meaning is a little different.
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spinyman"She could have invited him to come in"1. She had the ability to invite him to come in.
spinymanAnd yet, she could have invited you. I don't understand why she didn'tCorrect as shown. In this case I would not substitute might. With might I would add more to show the meaning 'maybe ... would', like this: