According to CASMA:

Organic shingles are made on a thick felt of wood and other cellulose fibres. Glass shingles are made on a mat of non-woven glass fibres, which are held together with a water-insoluble binder. Both shingle types contain asphalt, are covered with mineral granules.

Organic shingles:

• Are softer and easier to work with in cold temperature applications.
• Contain more asphalt.
• Absorb roof deck stresses better and are more tear-resistant than glass shingles.
• Are typically thicker than glass shingles and therefore hide minor imperfections in the roof deck or the old shingle layer.

Glass shingles:

• Are more resistant to heat, which may cause blisters to form on softer organic shingles on extremely hot roofs.
• Require the installation of an asphalt saturated felt underlayment.
• Roof assemblies with glass shingles have a higher fire resistance rating than roof assemblies with organic shingles.

If you have a well-ventilated, solidly constructed roof deck, and the shingle roof system is properly applied according to good roofing practices and the manufacturer's application instructions, generally both shingle types will perform well.