You're both right, and maybe a little bit wrong.
There's nothing wrong with listening to headphones, as long as you're not doing it in the many many situations where it's more important to be interacting with or just taking in your surroundings. Listening when you're, say, walking home from school is OK. Listening when you're in a social situation, or in class, not good. To be honest, you sound like you more or less have a handle on that, and maybe your Mom's being a little alarmist. But give her a break, that's what Moms do, they worry because they care about you. However, when you say 'other public places' it sounds like you might be doing it around your Mom. If this is the case, it obviously bothers her, so try to cut that out to reduce friction (and show common courtesy). If not, and you're just talking to her about how much/where you listen to 'phones, you should be able to do a better job communicating that you're exercising discretion. You seem articulate enough to do that.
Bottom line, your Mom will probably still worry some. So I guess my advice would be that a love of music is a great gift that will be with you your whole life, so that's great. But to make both of your lives easier, just try to be nice and reassure your Mom, and it will make both your lives a bit easier. Your lucky to have a Mom around who cares enough to make you miserable, ha ha ha.