What's the difference between mehadrin and non-mehadrin certifications.
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This can be a loaded question, but here goes:
Different kosher organizations can have different standards; the same organization can have two levels of standards, of which one might be "regular kosher" and another "mehadrin" (super-duper) kosher. E.g. in the page you linked, there are Rabbanut non-mehadrin, and Rabbanut mehadrin.
Often politics can play into things as well, unfortunately, but let's stick strictly with the food.
If I recall correctly from this lecture and elsewhere, here are a few matters that some "non-mehadrin" standards might accept, that "mehadrin" wouldn't. I'm not trying to demonize anyone here; again, as Rabbi Miller said, non-mehadrin is kosher. To us Americans, some of these are required by our usual standards (OU and the like); some aren't. I'll refrain from translating or explaining a lot of the jargon below, as I figure the question was intended for audiences with a stronger background.
I also heard, in the name of Rav Miller, that mehadrin vs. non-mehadrin relates to the issue of rov as it relates to checking on shochtim and whether animals were shechted properly. Again, to stress Shalom's point, this is not a matter of Kosher vs. non-Kosher (rov is a halachic principle which we can rely on), but a matter of stringency and perhaps for kashrut an issue of how you hold on the idea of timtum ha'lev...but that is another question.
In addition to what @Shalom said, the Rabbanut does not require a mashgiach temidi (permanent kashrus supervisor) in restaurants; only a yotzei v'nichnas (quasi-random inspections). This can be a major problem if the owner is not religious, especially for meat restaurants with basar shenis'alem min ha-ayin (meat that was not supervised).