As a preface to this entire question/answer, standards for things like BMI and BF% are based on averages. They work for most people, but there are also clear outliers. Of the actual outliers some people have naturally lithe bone structures while others have broad and thick bone structures. This will remain the case whether such an individual is large or small (in terms of weight, both muscle and fat). Therefore, this should at least be considered before trying to conform to a certain standard of BMI or BF%.
What's the difference between gaining weight by muscle gain and by fat gain, for an underweight person?
There is a big difference between gaining muscle mass and gaining fat mass. Regardless of a person's current size, they should seek to keep their bodyfat percentage within a healthy range. Now there isn't any need to obsessively reach a specific number with pinpoint accuracy, simply being within a healthy range is what should be sought after. I can't actually seem to find anything stating what that healthy range might be (please comment if someone knows), but for the sake of having actual figures a comfortable range would be between 10-20% bodyfat for men and between 15-25% for women.
Bodyfat and dietary fat play a big role in how your body manages its hormones, and sufficient bodyfat can of course keep a person from starving to death should such a situation arise. Outside of those two things however, bodyfat isn't very useful. Excess fat is just dead weight that hangs on the body and creates or contributes to a variety of unnecessary stressors.
Muscle mass supports itself, stabilizes and protects bones and joints, and grants a person a higher degree of mobility and strength. While there isn't any practical need for most people to build a great degree of muscle, it is beneficial to build any amount of muscle that a person is willing to work for (until perhaps you reach unnatural levels). There really just aren't any downsides to having more muscle mass than necessary.
... and that's where the answer to this question comes in. There are clear downsides to having more fat than necessary, but that isn't the case at all for muscle.
According to my belief, I should not go for muscle gain.
This is an incredibly unwise conclusion. You certainly don't have to reach for a Mr. Olympia title in the pursuit of building muscle, but if additional fat will be of little/no benefit to you and you need to gain weight, the acquisition of additional muscle mass should be your goal. Besides, a person will almost always and inevitably gain fat while gaining muscle if they are gaining overall weight. Keep in mind that the only way to build muscle is by providing a stimulus for your muscles to grow (resistance training being the gold standard). If you aren't doing this and you are gaining weight, it's probably just fat. Even if you are doing resistance training, if you aren't doing it seriously, it's probably still just fat.
I have a self made belief that muscle gain is for people who are already healthy... First... you should be a normal person with healthy weight. Am I right?
No, if anything an underweight person is in need of muscle mass the most. Again, this doesn't have to be anything crazy, but obtaining (at minimum) a basic level of strength is beneficial for your health. You don't want to become immobile when you get older just because you thought you were too cool for exercise as a younger person. As mentioned, you're also more prone to injury if you are underweight. This belief of yours is completely backwards.