What is an herb? What is a spice? Is there a difference?

Herbs are angiosperms that have little or no woody tissue. 

Herbs can be used for their aromatic (scent), savory (salty or spicy taste in foods), or medicinal purposes.

Spices are plant parts such as seeds, roots, bark and flowers, used to flavor food and beverages.

All spices aren't herbs and all herbs aren't spices.

A few common spices used to add flavor are:

Cinnamon-bark from a tree

Garlic- the bulb of a plant with little woody tissue

Cayenne pepper- a fruit from a short woody plant

Would any of these spices also be considered herbs?