I'm a junior web development tester (or "quality assurance specialist") with four months of testing experience lifetime. I'm working under the supervision of a female senior tester with about five years of experience. Her feedback to me is mostly useful and fair and I have learned quite a few things from her. However, the last few days she's been behaving a lot worse than normal:

Two days ago I did some administration for the both of us. Afterwards she said that my way of doing administration was correct, but she wanted me to do the administration her way next time. I said that yes, her way worked too, but I explained why I thought that my way was easier and faster, yet equally good. She said that she wanted me to do the administration her way because she was used to that. I gave in because she's my supervisor.

We usually divide up the tasks between us and work mostly independently. Today I saw her make a somewhat serious administrative error regarding a task that was put on her name and was her responsibility. I wasn't explicitly double-checking her work, I just happened to see it. There's no "other way of doing things" this time, it was just plain wrong, but it certainly could be repaired in say a minute so that it was correct and still "her way of doing administration".

So I sent her an email: "hello [her name], there's an error here, can you look at it? Kind regards, [my name]."

(For context, we live in a very egalitarian country and pointing out your supervisor's errors is accepted and relatively common here. That being said, maybe I should have worded that email more politely or, let's say, carefully.)

She sent me an email back saying something like "hello [my name], thanks for the comment. I'm busy right now. I know you're autistic and really like having proper administration but I can't always accommodate that. Also, I think your comment is very rude and nosy." (At the end of the day I would see that she still hadn't fixed her error.)

Yes, I have Asperger's syndrome and she knows that. Yes, I've been too nitpicky in the past (though I'm improving on that). Yet I don't like that rather than admitting her error, she blamed me for pointing out her error. I know I'm biased here but I honestly think that a neutral observer would say that in this case I'm not being overly critical and the administrative error should be fixed.

Also, I think it's hypocritical that she says "do things the way I'm used to, despite your way also being correct and despite your reasons why you say your way is better." Yet when she makes a genuine error and I point that out, she plays the victim, pressure flips me, plays on my insecurities ("you're an annoying detail-obsessed nitpicker and everyone is merely tolerating you"), ignores my actual point and then doesn't fix her error. I know she's my supervisor which somewhat justifies that she gives me feedback while not accepting feedback from me, but still.

More context: while she's absolutely better at some aspects of the job than I am, I'm better than her in test automation (she's primarily a manual tester) and some other areas. A senior developer recently told me one-on-one that he thought I was a better tester than my supervisor was. I don't actually think that I'm better than my supervisor because I lack experience, but in say a year's time I could be. So this might contributed to her reaction.

I had a tight deadline to make and no time to talk to her, so I swallowed my anger and emailed her back "thanks for the feedback, to be absolutely sure that I understand you correctly: did you take offense at the wording or at the contents of my email? To put it differently, do you think I should have been nicer/more polite or do you think that I simply shouldn't interfere with your tasks?" She didn't reply to that, I kept working and eventually went home.

So, what should I do about this current situation? Should I approach her and try to talk it out or should I simply not bring it up again and only talk about it if she starts the discussion? Should I request a meeting with my team lead and bring this up, or would that be making too big an issue out of this?

If we do talk about it, I'll try to find out if there's some point that I'm actually missing. If it turns out that she actually didn't have a good reason to be so hostile: should I stand my ground or should I apologize for something that I think is more her fault than mine?

tl;dr: I told my supervisor she made a mistake, she reacted in a very hostile way. Should I apologize for the sake of my job, should I think "she's normally much more reasonable, so let's ignore her outburst" or should I contact my team lead (who is higher than either of us in the company's hierarchy)?