Interesting topic.
From what I read, it seems one is spiritual and the other materialistic.:
Idealism believes in what is beyond the senses, the spiritual and mental world and a search for Truth, Beauty and Justice. It seems to be a search of perfection. A philosophical approach based on the tenet that there is one reality which is worth knowing.
Emphasis of studying is on reasoning and concepts with a Teacher to guide.
Quote: "Plato also believed that the soul is fully formed prior to birth and is perfect and at one with the Universal Being. The birth process checks this perfection, so education requires bringing latent ideas (fully formed concepts) to consciousness."
In Realism basically one believes in the objective world which the senses can cognize. Rational thinking and analysis is the purpose of human kind, expounded by Aristotle.
It speaks of systematic study and application of knowledge to understand, basically experimental learning.
Quote: "Reality exists independent of the human mind. The ultimate reality is the world of physical objects. "
The other two school of thoughts are Pragmatism and Existentialism.