Natural man vs carnal man

Sounds right to me. Here is what Robert Alexander Webb, “Man, Natural,” The International Standard Bible Encyclopaedia has to say on this topic. The Bible says to put off the old man and put on the new. Ephesians 4:22

The Natural Man

The “natural man” is the “old man,” the “outward man,” the “carnal man”—man as he is by nature, as he is firstborn, contradistinguished to man as he is changed by the Spirit, as he is second-born or regenerated. There is an “old” life, an “outward” life, a “carnal” life, a “natural” life, as contrasted with the “new” life, the “inward” life, the “spiritual” life, the “gracious” life. The “natural man” is a bold and vivid personification of that depraved nature which we inherit from Adam fallen, the source and seat of all actual and personal transgressions.

II. Theological Meaning.—We know what we mean by the nature of the lion, by the nature of the lamb. We are using perfectly comprehensible language when we speak of the lion as naturally fierce, and of the lamb when we say he is naturally gentle. We have reference to the dominant dispositions of these animals, that resultant of their qualities which defines their character and spontaneity. So we are perfectly plain when we say that man is naturally sinful. We are but saying that sinfulness is to man what fierceness is to the lion, what gentleness is to the lamb. The “natural man” is a figure of speech for that sinful human nature, common to us all. It is equivalent to the theological phrases: the “sinful inclination,” the “evil disposition,” the “apostate will,” “original sin,” “native depravity.” It manifests itself in the understanding as blindness, in the heart as hardness, in the will as obstinacy. 

NEW MAN: Generally described, the “new man” is man as he becomes under the transforming power of the Holy Spirit, contrasted with man as he is by nature. The phrase has (1) its Bib., and (2) its theological, meanings.

I. Biblical Meaning.—There are four Bib. contrasts which must be considered as opposites: (1) the “old man” (palaiós anthrōpos) and the “new man” (neos or kainos anthrōpos); (2) the “outward man” (éxō-anthrōpos) and the “inward man” (ésō-anthrōpos); (3) the “carnal man” (sarkikós anthrōpos) and the “spiritual man” (pneumatikós anthrōpos); (4) the “natural man” (psuchikós anthrōpos) and the “spiritual man” (pneumatikós anthrōpos). These are not four different sorts of men, but four different sorts of man. Take up these antitheses in their reverse order, so as to arrive at some clear and impressive conception of what the Bib. writer means by the “new man.”

1. The Spiritual Man

The “spiritual man” is a designation given in opposition to the “carnal man” and to the “natural man” (Rom 8:1–14; 1 Cor 2:15; 3:1, 3, 4; 2:14; 3:11; 14:37; 15:46; Gal 6:1; Eph 2:3). All three of these terms are personifications of human nature. The “carnal man” is human nature viewed as ruled and dominated by sensual appetites and fleshly desires—as energized by those impulses which have close association with the bodily affections. The “natural man” is human nature ruled and dominated by unsanctified reason—those higher powers of the soul not yet influenced by Divine grace.

The “spiritual man” is this same human nature after it has been seized upon and interpenetrated and determined by the Holy Spirit. The word “spiritual” is sometimes used in a poetic and idealistic sense, as when we speak of the spirituality of beauty; sometimes in a metaphysical sense, as when we speak of the spirituality of the soul; but in its prevalent Bib. and evangelical sense it is an adj. with the Holy Spirit as its noun-form. The spiritual life is that life of which the Holy Spirit is the author and preserver; and the “spiritual man” is that nature or character in man which the Holy Spirit originates, preserves, determines, disciplines, sanctifies and glorifies.

Robert Alexander Webb, “Man, Natural,”  The International Standard Bible Encyclopaedia

I think Tozer had it right. 

You can always test the quality of religious teaching by the enthusiastic reception it receives from unsaved men. If the natural man receives it enthusiastically, it is not of the Spirit of God. Paul says plainly that the natural man cannot know spiritual things. To him, spiritual things are plain foolishness (see 1 Cor. 2:14).

There is a type of religious teaching understood, received by and perfectly logical to the natural man. But the natural man does not know that which is of the Spirit of God. He does not have the faculty to receive it.

The natural man is of this world. He may be in perfect health and have an IQ of 180. He may be as handsome as a Greek statue or, if a woman, a perfect example of fine womanhood. Or he might be a perfect example of the young American. The natural man, though he is in this state, is unblessed and out of grace.

Contrary to the natural man is the spiritual man. This is the Christian who is mature in his faith, who is led, taught and controlled by the Holy Spirit, and to whom the Spirit of God can speak.

Then there is the carnal man. The carnal man is the immature Christian. He is no longer a natural man, for he has been renewed by the grace of God and is in a state of grace, but he is not spiritual. He is halfway in between the two. He has been regenerated but is not advancing in his spiritual life. He is not influenced or led by the Holy Spirit but rather is controlled by his lower nature.

Of the three types, it is the spiritual man who is living the crucified life. He is indwelt, led, taught, influenced and controlled by the Holy Spirit.

A. W. Tozer, The Crucified Life: How to Live out a Deeper Christian Experience

The reason I started this thread was to clarify the difference between Calvinist's claim that the natural man cannot believe the gospel, while the carnal man had to be regenerated first to believe the gospel only to live carnally.
I was seeking a distinction between the two as to why one who is natural cannot believe, yet a believer could behave carnally.

Neither can the carnal man understand the things of God:
1Cor 3:1  And I, brethren, could not speak unto you as unto spiritual, but as unto carnal, even as unto babes in Christ. 
1Cor 3:2  I have fed you with milk, and not with meat: for hitherto ye were not able to bear it, neither yet now are ye able. 

Thus the OP; the natural man and the carnal man, both being not spiritual, cannot understand the things of God; so what is the difference?

In my opinion, there is a common mistake in hermeneutics when dealing with this issue. 

While there are differences, there is no such thing as SEPARATING them. They both have their own qualities and attributes but they "FORM" the whole. Which is a mistake common in Dispensational theologies. "Rightly dividing" the Scriptures is a process of identifying how the entirety of the Scriptures FIT TOGETHER to form the WHOLE. 

Too often a "DIVIDE" is made without any thoughts of how they "FIT TOGETHER". 

Notice in this example, pay close attention to what Paul said. 

He uses the words "babies". Carnally..... FLESHLY.... They were not babies. However, Spiritually.... they were very young. Inexperienced. This lack of experience was affecting the WHOLE of their existence. 

You are absolutely correct. 

Spiritual babies don't understand everything they see nor hear. So there are levels to understanding that the Calvinist ignores in the discussion. 

Which is observable in the Gospel message. It gets the attention to the hearer. The hearer MUST deal with the information they are presented. Paul even said that Satan had BLINDED THE MINDS of SOME...... again. Not all. So they would not even begin the journey to discovery. 

The GOSPEL LEADS ALL MEN...... ALL MEN....

Some further than others. 

I'm totally against Calvinism and only trying to point out their error.