The way I see it, there isn’t much difference in meaning. The main differences are grammatical and formality. “How come” is more informal than “why,” and you would set up the sentence differently depending on which one you use. Say I said “The sky is blue.” “How come” is simply added to the beginning of the statement to turn it into the question — “How come the sky is blue?” With “why,” you have to rearrange the word order — “Why is the sky blue?” If you are in a professional situation, at work, sending an email, stuff like that, I would stick to “why” because “How come” has a much more informal or childish feel to it and isn’t as proper.