What are the types of charcoal pencils and how do they differ from each other?
These charcoal pencils are just a demonstration of the styles of pencil you might encounter.

Wooden Coated Charcoal

The first pencil, labeled "soft" is a softer charcoal pencil, coated in wood. Wooden charcoal pencils are generally cheaper and require a sharp blade (ex exacto knife) to sharpen them. They would not easily sharpen in a traditional pencil sharpener as the charcoal is so soft that a regular sharpener would grind and break the charcoal during the sharpening process.

Paper Coated Charcoal

The second pencil is coated in paper. Paper coated pencils are generally more expensive, but provide the ease of being able to reveal more of the charcoal by simply pulling the string and unraveling the paper rather than sharpening the pencil with a blade. The 600-H identification on the pencil is just Prang's model number for that pencil and means nothing with respect to the charcoal grade.

White Charcoal

The 558 is General's line of white charcoal pencil and, like the 600-H is proprietary to the company and has no real meaning with respect to the charcoal. 

White charcoal is obviously different from black charcoal but is used in much the same way, but in reverse. White charcoal is a great way to come back after the work is finished to really bring out highlights and shines to be a "whiter than white" which brings contrast and boldness to an image.

White charcoal really isn't charcoal at all, but a type of pastel or Conté, often chalky and meant to be used with charcoal. It can be made from mixing white pigment with some type of binder, but the exact makings are generally proprietary and not disclosed.