I struggle to see a difference between comparing a large number of anecdotes on Twitter and comparing data gathered by professional agencies. I'm not sure there's a difference beyond professional authority, but one is usually considered evidence while the other isn't. That is, people are quick to say "anecdotes aren't evidence", but also are quick to trust that research articles contain genuine truth. But what is a research article? Isn't it really, in many studies, "I/we did x" (which is an anecdote) or "I/we gave x to experimental group y and not control group z" (which is an anecdote)?
Granted these studies are meticulously in-depth anecdotes, but they are essentially anecdotes. That is, they are one person, or a small group of people, telling a story about something they did or saw or experienced. Case studies are the most obvious examples of this -- they are, essentially, incredibly in-depth anecdotes.
So what exactly is meant by "anecdotes aren't evidence"? Is there a threshold of information one must pass before their anecdotal evidence becomes "real" evidence? Even supposing they presented their statistics in extreme depth, isn't the presentation of those numbers itself anecdotal? Phrased in a different way (an obnoxious way), a table in a medical study might look something like: "We gave Billy this treatment and he improved, we gave Johnny this treatment and he improved, we gave Sarah this treatment and she improved..." etc., which is entirely anecdotal. A physics experiment is subject to the same classification.
If every publication in empirical research can be reduced to an anecdote while still retaining its essential features, why are anecdotes not evidence?