Would a high stack exchange rep make a difference in a job interview?

I have seen Math.SE and MathOverflow activities mentioned by both grad students and faculty in connection with various applications and promotion-related materials. I don't see this as anything to scoff at; personally I find it intriguing and consider looking up someone's SE profile(s) a great way to get some insight into their mathematical personality (note that there can be a potential downside if said personality turns out to be uninteresting or unimpressive). With that said, I think this is only worth mentioning on your CV if it represents a major activity, say if you have been active for at least 1-2 years and have dozens of answers including a good number of highly voted ones. I wouldn't specify the reputation points as that seems like a silly thing to (admit to :-) ) be concerned about. And there can be a better place to mention it, e.g., in a research/teaching/personal statement.

As for mentioning this in an interview, like many other things this can be a good idea if done tactfully (in a no-big-deal, casual sort of way that shows a sense of proportion), or a horrible idea if done untactfully (saying it and expecting fireworks or a round of applause).

Finally, I wanted to mention that I noticed from questions such as this one that there is a common belief in the Grand Principle that for academic (and specifically math) jobs one should hide any information about oneself that attests to one being interested in anything other than doing math research all day, every day, and that any decision about what to put on one's CV or what to mention in an interview/personal statement should be considered in the light of this Grand Principle. I can't say if this is true or false as a general rule, but it's categorically false for me: personally I'm delighted if a job candidate tells me they are interested in archery, or chess, or surfing, or playing piano, or whiling away their time answering elementary questions on Mathematics StackExchange. Think about it this way: given that I already know the level of research they do (which presumably must be pretty good if I'm interviewing them), any evidence that they have other unusual skills or abilities outside of math research can only raise them in my estimation. Besides, when you hire someone you get a complete package, mathematician/academic+human being, and personally I rather like being surrounded by interesting human beings and not just paper-writing robots. But maybe I'm weird and it's just me who thinks this way, so handle this advice with proper caution.

As a committee member/voting faculty member (at R1 place) on relevant things, etc., I'd say that mention of "high reputation" would seem childish, but as "outreach", activity on such sites is a plus. It is "outreach", which is good, if not super-glam. In the future, it may be more than "outreach" (which is not exactly the right descriptor, anyway), but for the moment, this is a reasonably modest description... and does resonate with current "allowable" categories of "good things" on CVs and such. So, no, don't tell what "hats" you got around Xmas, nor "medals" or whatever they are, but perhaps mention _outreach_activity_. (Five or ten years from now, I imagine the descriptors and the ground-rules will be different...)

I looked at your network profile, and knowing your MSE rep would not make a difference for a job interview at a research school. (I'm not belittling--my MSE rep is about the same as yours! (currently in the 800s)) For a liberal arts/teaching school, knowing you are active on MSE may help a little. So sure you can mention it as supporting evidence of enjoying to answer students math questions, and you could even mention in a teaching statement if it supports your claims. However, I would definitely not mention rep anywhere.

I vaguelly recall a similar question on MO or MSE (there have been many on other SE sites), where someone said just to mention it on your CV if you're a moderator, however I can't find such a post, so I may be imagining things. (I personally wouldn't find it offputting to see something like "MathStackExchange, active contributor" on a CV if your profile used your real name and you are pretty active. Note: I don't consider myself as active on MSE.)