Yes, there is a difference, and it's more than just formality. "Why?" is very direct. It's a straightforward request for a reason. "How come?" is a bit more general. It means "Tell me about the circumstances which led to this situation ". It means "Hmm, there's something I don't quite understand here. Can you give me any background?" or "Oh! That's surprising. How did that happen?" Let's say your friend, who speaks excellent English, took an English test yesterday. Today, he tells you that he failed his test. In this situation, you might not say "Why did you fail?". If you asked that, it might sound as though he did something deliberate to fail (which wouldn't be likely), or it might sound as though you're asking a very obvious question ( he failed because his work wasn't up to standard). A better question would be "How come you failed?". This shows that you are surprised at this outcome, and you are asking how this came about.

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.

I would say that "How come?" is less formal than "Why?". In an interview or business meeting, for example, you would ask "Why?" rather than "How come?" as it translates a more respectful or traditional in tone.

There's a big difference; you can't use 'how come' in any situation. why = tell me the reason how come = explain how this thing came about so you can't say 'how come you bought a car?' because it was your decision, not something that happened. but you can say 'how come you didn't go?' - it means the person wanted to go, but something happened to stop him. ('why didn't you go?' is just a normal question, asking for his reason)

"Why?" is straightforward, direct. It doesn't suggest that the situation is surprising. It is just a request for an answer or an explanation. "How come?" is specialized. It implies that you find the situation puzzling or surprising. It often conveys a sense of annoyance, or even skepticism. Roughly, you can almost say that "how come?" is short for "if that's really true, then how come..." "Why should I include stocks in my investment portfolio?" "Historically, over the long term, they have made more money than bonds or cash." "If stocks are such a great investment, then how come I have less money now than I did at the beginning of the year?" "Because eleven months isn't the long term." "Why is it colder in winter than in summer?" "Because the sun is shining on the earth at a lower angle, and the same amount of sunlight gets stretched out over a larger area of land." "If that's so, then how come it is 16° today in November, when it was only 14° back in August?" "Because the angle of the sun only one of many factors that influence the temperature." "Why is the long e sound in 'receive' spelled 'e-i' while the same sound in 'believe' is spelled 'i-e?'" "Because the rule is: 'I before e, Except after c, Or when sounded like 'a' As in 'neighbor' and 'weigh,' Except 'seize', 'inveigle,' 'either,' 'Weird,' 'leisure,' 'neither.'" "Then how come 'science' is spelled with an 'i-e' after 'c?'" "Because no spelling rules in English can be trusted."