It's actually been scientifically proven that men have a difficult time remembering dates, I'll find the study if you really want but I promise it's true.
Look, I don't mean this as an insult but you're not even 18 yet, you're still in high school, there are bigger problems to look at than if your boyfriend remembers your birthday. It was a mistake, we all make them. Hell I can't even remember what years my parents were born, and I keep on forgetting what day exactly in November my boyfriend's birthday is. And he constantly needs to be reminded of mine. We're hopeless. It happens.
Both of you are going to forget a lot of things and if you hold that against him every time he messes up or forgets something it's going to be a miserable relationship with you constantly telling him what a failure he is because you can't let anything go. Not to mention it's barely been over a half of a year that you guys have been dating. You should take a deep breath and just let it go. Occasionally my boyfriend says stupid things when he's not thinking, we're not just boyfriend and girlfriend we're also best friends (the only way any long term relationship works) so sometimes we forget that we need to treat each other differently than our friends we we both let stuff slip that isn't acceptable in a relationship.
But ya know what? We forget it because we know we don't mean it and that the ability to let that kind of thing slip out proves just how close we are as friends. If you can't let this go you'll never progress in the relationship because you'll just cling on to every little thing he does wrong and horde it and eventually it'll all explode. You just need to sit down, write it out, talking it out to a pet or a pillow, dance it out, sing it out, whatever it takes to just let it go. Because he is sorry, and he does care deeply about you and he won't make that mistake again. You're being too hard on him and if you treat him like this then he has the right to dish it right back when you mess up or forget something.