Fiberglass and mineral wool batts are the most common insulation products worldwide.
Mineral wool is widely used in Europe and a little all over the world, while fiberglass is mostly used in the North America.
Both products are safe (though irritant to skin, during installation), cheap and lightweight.
Overall, mineral wool batts are a much better product than fiberglass: mineral wool is easier to install correctly, more water resistant and with a better R-value.
Fiberglass is a non-organic product made from melted glass spun into a mat of fine fibers. Mineral wool is also a non-organic product, made from volcanic sand melted at high temperatures and spun into a mat.
Installation cares & Performance
Some people claim that fiberglass fibers, when inhaled, are carcinogenic. But institutions as the American Lung Association state otherwise. Be careful, anyway. Direct contact with fiberglass materials (or exposure to airborne fiberglass dust) may indeed irritate the skin, eyes, nose and throat.
Fiberglass and mineral wool performance depends largely on proper installation; batts must fit the cavity or the framing elements closely, without voids, gaps, splits, compression, indentation…
And that's where mineral wooll batts are largely advantageous, compared to fiberglass.
Fiberglass have to be hung and be carefully stapled into place to avoids gaps and not to slump.
It's hard to install fiberglas properly. It doesn't retain its shape.
Mineral wool is denser, and doesn't drape or fold easily; it retains its shape and fits into studs correctly, avoiding convection, drafts or stapling.
In other words: the performance of fiberglass insulation is easily degraded by poor installation; it's happening all the time, contrary to what happens to mineral wool.
The R-value of common fiberglass batts sold in the U.S. is lower than in Canada. There is differences that you should take into consideration.
The R-value of mineral wool is higher than that of fiberglass. In the US, mineral wool batts for 2x4 stud walls comes in R-15, while common fiberglass comes in R-11 or R-13 (excepcionally R-15); similarly, batts for 2x6 walls comes in R-23, while common fiberglass comes in R-19 (special ordered batts may come in R-21).
Higher density fiberglass and mineral wool batts allow higher insulation (R-values) for the same space. There is high-dense mineral wool batts especially designed to be used as insulating sheathing - to cover the surface of walls, preventing thermal bridging.
Water resistance & Fire Resistance
Though fiberglass has a good water-resistance, it doesn't shedd water and it isn't fireproof - contrary to mineral wool.
In the North America, see NAIMA. See: NAIMA members
Mineral wool manufacturer for the residential market, in the U.S.: Roxul.
To find contractors in the North America:
Insulation Contractors Association of America Contractor Locator.
Blow-in-Blanket Contractors Association Contractor Locator (North America)
Resnet insulation contractors (the search demands your Zip code and the type of contractor you are looking for)
For non-american associations, see the NAIMA list of fiberglass and rockwool associations outside USA.
To find European mineral wool companies (and their dealers, contractors and other insulation professionals) see: Eurima Members.