Here's how the great classicist and Thomist Josef Pieper explains them in his masterful Leisure, the Basis of Culture:
The middle Ages drew a distinction between the understanding as ratio and the understanding as intellectus. Ratio is the power of discursive, logical thought, of searching and of examination, of abstraction, of definition and drawing conclusions. Intellectus, on the other hand, is the name for the understanding insofar as it is the capacity of simplex intuitus, of that simple vision to which truth offers itself like a landscape to the eye. The faculty of mind, man's knowledge, is both these things in one, according to antiquity and the Middle Ages, simultaneously ratio and intellectus; and the process of knowing is the action of the two together. The mode of discursive thought is accompanied and impregnated by an effortless awareness, the contemplative vision of the intellectus, which is not active but passive, or rather receptive, the activity of the soul in which it conceives that which it sees.
Ratio is discursive reason: induction and deduction. Modernists hold it to be the only valid kind of knowing.
2. Claims can be demonstrated logically; reasons can be given to defend them
3. Is public, indirect/sequential (discursive; step-by-step, chronological--chronos.)
4. Takes effort on the part of the knower to compare, examine, relate, distinguish.
1. Is a type of knowing which is not sequential, but all-at-once and therefore "timeless" (kairos.)
2. is direct/immediate and private/personal
3. is given, not earned; in-spired, not induced. The knower is passive, receptive.
4. Medievals saw it as the highest form of knowledge, more typical of the way God and the angels know, than of men; nevertheless, insofar as man is spiritual, he has the ability to know in a super-human, divine way.
5. Focuses on real things, which are both spiritual and material.
6. Contemplation, meditation, visions;
7. Intellectus is the way Paul knew some things…for example,
2 Cor. 12:1-4
“Although there is nothing to be gained, I will go on to visions and revelations from the Lord. I know a man in Christ who fourteen years ago was caught up to the third heaven. Whether it was in the body or out of the body I do not know—God knows. And I know that this man—whether in the body or apart from the body I do not know, but God knows—was caught up to paradise. He heard inexpressible things, things that man is not permitted to tell…”
I Cor. 14: 1-2, 18-19
“Follow the way of love, and eagerly desire spiritual gifts, especially the gift of prophecy. For anyone who speaks in a tongue does not speak to men but to God. Indeed no one understands him, he utters mysteries with his spirit… I thank God that I speak in tongues more than all of you, but in the church I would rather speak five intelligible words ito instruct others than ten thousand words in a tongue."
I explain ratio and intellectus to my Intro class by presenting them a live rose. I ask, "How can I know this rose?"
First, I recognize it as "rose," and not as "peanut" or "shoe."
That doesn't take any effort on my part; I abstract the form of rose in an immediate intuition.
But then I can know the rose qua rose. That is, I can let it act on me, receiving its shape and scent and color in a timeless moment of contemplation. I am passive before it, receiving it "into me" as a "gift" worthy of praise, thanksgiving or imitation. That is to know it by intellectus.
"But I can also act on the rose; I can analyze it. I count the number of leaves and thorns; describe the shape and color. (I pull off the petals, one by one, and let them fall into a pile.) I can determine the number of petals. I can classify this rose as Damask or Noisette or Hybrid tea or polyantha or ? I can breed new roses by removing all the petals and stamens of this rose and introducing the pollen of another rose. I can use rose hips to make a tea high in vitamin C. This is to know it by ratio."
The danger of our time is to live in the wake of the either-or, and to EITHER insist with the moderns that knowledge is ratio, OR insist with the postmoderns that it is NOT ratio, and therefore intellectus. As a premodernist, I affirm both-ands. It is proper for angels to know only by intellectus; it is proper for computers to know only by ratio; but we have been made "a little lower than the angels." We exist in time and space, and yet we are able to transcend time and space. Thus, human beings alone have the ability to know both through ratio and intellectus.