What you are measuring...

I was trained as a physicist. When we made a single measurement, of an event or a whatever, we have a sense of the measurement error, what we called systematic error. It usually depended on the quality of our apparatus and its limits. In the case of rare events, we might even have a sense of a Poisson process and so say that we saw an event at time T. In any case, whatever it was that we were measuring was as real as anything, not an artifact.

When we combined measurements, a combination of the data, that is we were being statistical, we might expect to get something like a gaussian with a mean and standard deviation (or some other expected or suprising distribution). The more events we had, the better known were those statistics. And we believed that those statistics were referring to something as real as anything--say the mass of the particle and its lifetime (the inverse of the standard deviation) with an unavoidable systematic error that affected our statement of the standard deviation and perhaps that of the mean.

If you wondered whether the mean were different from zero, you wondered whether some real thing were different from zero--say the mass of the tau neutrino. If you wondered whether the mean of two different measured quantities were different, you were wondering whether the mass of the K particles were different. You would take the width or standard deviation seriously, because in fact there was surely some systematic error and there was an intrinsic width of the measured quantity, its lifetime, but still the mean, the mass was something real, and you would quote the mean with an error and the standard deviation with an error (systematic and statistical)

In social science studies, as far as I can tell (and I have become acutely aware of this only recently), again we make a single measurement and have sense of measurement error (if that makes sense in this context: sometimes it may be a matter whether people are reliable reporters in a survey, whether the data is dirty,...).  Again, we might say that whatever we are measuring or surveying is as real as anything.

Again, "When we combined measurements, a combination of the data, that is we were being statistical, we might expect to get something like a gaussian with a mean and standard deviation (or some other expected or suprising distribution). The more events we had, the better known were those statistics."  And we act as if those statistics were referring to something as real as anything--say the average height of a population. But almost always there is no reason to take those statistics as real, they were artifacts of our combination and we had no theory that gave that number a deep reality--or so is my observation of actual practice. And of course there might well be "an unavoidable systematic error [ore measurement] that affected our statement of the standard deviation and perhaps that of the mean."

If you wondered whether the mean were different from zero, you would check the power of your statistic, you would see how well measured it was (standard error) and so you might get a good sense of whether the mean were different from zero. But, there was nothing real about it in the sense that a particle mass were real. It was a statistical measure of that artifact, the height of a population. Presumably the width is substantial but not overwhelming, but it shows the dispersion of heights.

However, say you wanted to check whether the difference in heights of two populations were significant. Surely you can do much as the physicists would do, and see if the difference of the means were statistically significant (and say that the systematic or measurement errors were not important). But say as well that the distributions overlapped substantially. You could surely say something about whether the means were different. But, I would find it hard to take such a statement very seriously, since the distributions overlap so much and so any problems in the distributions would make me skeptical that the measured difference were credible in actuality.