There is a distinction.
The Spirit Himself witnesses with our spirit that we are children of God
(Rom. 8:16; cf. John 1:12).
The Spirit witnesses to our most elementary relationship with God, that is, that we are His children; it does not witness that we are His sons or His heirs.
The fact that the "begetting" Father wants His children to grow unto maturity is presented clearly in Ephesians 4. Paul says that the Father has predestinated us unto sonship (not unto salvation) through Jesus Christ (1:5).
God predestinated us not to be His children, remaining in the initial stage, the stage of immaturity, but to be His sons, attaining to the final stage, the stage of maturity.
This is the background for Paul's exhortation in chapter four of Ephesians, where he says,
Until we all arrive at the oneness of the faith and of the full knowledge of the Son of God, at a full-grown man, at the measure of the stature of the fullness of Christ (v. 13).
In order to arrive at this degree of maturity, we should be “no longer little children tossed by waves and carried about by every wind of teaching in the sleight of men, in craftiness with a view to a system of error” (v. 14). For this, we must hold to truth in love that “we may grow up into Him in all things, who is the Head, Christ” (v. 15). By means of such growth, the children of God become the sons of God for the fulfillment of the God's purpose.
Paul's burden for the believers' growth to maturity permeates 1 Corinthians.
I, brothers, was not able to speak to you as to spiritual men,” he
tells the saints at Corinth, “but as to fleshy, as to infants in
Christ” (3:1). And “Brothers, do not be children in your
understanding…in your understanding be full-grown” (14:20).
As the result of their growth in the divine life, the sons of God, in contrast to the children of God, are mature, full-grown, in their understanding.
The fact that sonship, the making of believers into mature sons of God, is an organic process can be seen in numerous places in the New Testament.
Brendan Byrne points out the tension that exists between the present and future aspects of υίοθεσία in Romans 8 (if υίοθεσία is viewed as adoption). This can be resolved by an understanding of “a real but hidden status of υίοθεσία in the present, attested by the Spirit (vv. 15-16) and the public revelation of this status at the time when believers will share the bodily resurrection of the Firstborn Son of God (v. 29; cf. Phil. 3:20 1)." (Byrne, Brendan. Rev. of "Adoption as Sons of God—an Exegetical Investigation into the Background of υίοθεσία in the Pauline Corpus." Ed. James M. Scott. Journal of
Theological Studies 44 (April 1993)
With regards to Romans:
There is a progression to become sons of God through salvation in life (5:10), which, according to Romans, includes sanctification, renewing, transformation, conformation, and glorification (6:19; 12:2; 8:29-30).
The more we are sanctified, renewed, transformed, and conformed, the more we become sons of God, awaiting "the freedom of the glory of the children of God" (v. 21). This glory is related to the redemption, the transfiguration, of our body, which is the full sonship (v. 23).
The word heir κληρονόμος indicates maturity. Sons of God are heirs (Gal. 4:7; cf. Rom. 8:17). Son is particularly significant in dealing with inheritance. In ancient times inheritance laws privileged the son who inherited everything when he came of age. When the believers attain to the maturity in life of being mature sons of God, they are qualified to inherit all that God is and has for them.