In his article, Martin Cothran compares and contrasts the most basic positions of two ancient philosophers.  Plato, he says, emphasized the spiritual realm and felt that physical objects were reflections of that spirit world.  Aristotle believed the essence of things resided IN the things themselves.  They were real physical forms inhabited by spirit. Why does this matter to a busy homeschool mom?  Lets put it in practical terms.  We choose to teach either from the concrete to the abstract or vice versa.  It is important to know how you feel about this and choose materials and methods accordingly. 

Montessori moves from the concrete to the abstract, hence the emphasis on physical manipulation early on.  She believed that small children relate best to what they can see, smell, hear and touch.  They begin with the physical world and move into the spiritual or abstract.  From things to ideas.  

Waldorf does things the other way around.  The belief in reincarnation is largely responsible for this inverse order.  Steiner believed that people were essentially spirits who occasionally reincarnated into the physical world.  Children, believed to be newly arrived from this spiritual realm, were more familiar with ideas and images and had to be slowly  acclimated to physical life.  Hence the reluctance to introduce text and data at an early age. 

Both methods intend to move from the known to the unknown. They simply have a different position on what constitutes "known" for a small child.   Consequently the methods, while they enjoy some overlap, are essentially the inverse of one another.  While materials for both methods are now widely available and often exceptionally attractive they do offer completely contrasting approaches.  This article may help you make choices among them and determine very practical things such as whether you will proceed from the whole to the parts or the parts to the whole.  In the end, most curricula can be divided into one of those two camps.