What's the difference between mehadrin and non-mehadrin certifications.
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.
- "Heter mechira" for Shemita
- Wholesale separation of Trumah (as opposed to doing it again smaller-scale); several decades ago, a few rabbis even employed a heter of R' Shlomo Zalman Auerbach's whereby the trumas maaser becomes batel; not sure this is still around anymore.
- How much sifting is required of flour to avoid bugs.
- Relying on USDA or similar supervision to count as Chalav Yisrael (what we call "chalav stam") for powdered (based on the ruling on R' Tzvi Pesach Frank -- but not liquid milk); if you're in Israel this is only an issue for imported products, as most domestic dairy production involves Jews anyhow. With regards to liquid milk, there was an OU mp3 which said some Israeli hechshers are open to the idea of certifying a dairy production as Chalav Yisrael if an overseer is simply watching on video (as opposed to being physically present).
- In some cases, certain forms of animal-derived kosher gelatin.
- Salting the internal cavity of a chicken without splitting it open.
- In the past, freezing meat from South America and only soaking/salting it upon arrival in Israel ("basar kafu"). Though I've heard they now soak/salt it in South America within 72 hours of slaughter anyhow, so this may be moot.
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).
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.