I'd like to have a go at explaining the difference but I don't have any sermons to recommend. I've heard the idea (about maturity) which you mention, but is it justified, if we believe that to be born again brings us into a new relationship with God spiritually, as a new creature, in which the need to grow and mature is implicit? I wonder. The Holy Spirit in us cries 'Father'. This is the most compelling way a believer knows he has been born again.
So, regarding the difference between 'child' and 'son', I've noticed a few scriptures which help me.
This is the KJV.
Eph 2:2 Wherein in time past ye walked according to the course of this world, according to the prince of the power of the air, the spirit that now worketh in the children of disobedience: 3 Among whom also we all had our conversation in times past in the lusts of our flesh, fulfilling the desires of the flesh and of the mind; and were by nature the children of wrath, even as others.'
In v2, 'disobedience' means 'unbelief'. It implies (almost) that these 'children', have chosen not to believe (because they had heard the word and refused it; it is the word that creates faith), whereas, 'were BY NATURE children of wrath', covers us all, before we heard the word of God (to open our ears to hear, thus creating faith: Rom 10:17).
In other words, 'children' is a general term for 'offspring of God' (Acts 17:28, Luke 3:38), but we were all 'in Adam' when he died through disobedience, and thus we became 'children of wrath' by default.
But Christ came to bear the 'wrath of God' FOR us, and some 'children of wrath' - that is, those upon whom God's wrath would have fallen had not Christ died for us AND we HAVE BELIEVED in Him - are no longer 'children of wrath'; through faith we are no longer 'children of wrath': 1 Thess 1:10 And to wait for his Son from heaven, whom he raised from the dead, [even] Jesus, which delivered us from the wrath to come.
Jesus said: Matthew 11:19 The Son of man came eating and drinking, and they say, Behold a man gluttonous, and a winebibber, a friend of publicans and sinners. But wisdom is justified of her children.
So, what are wisdom's 'children'? This has puzzled me for years! But, I now think (apart from the context of the particular verse, which is another discussion), that they are the fruit of wisdom. For instance, if the fear of the Lord is the beginning of wisdom, then that right kind of fear begets a kind of godliness, which is a SIGN of the wisdom it follows on from.
I realise the last example seems to be the odd one out, but the more you read the New Testament closely, if you check up on the times child appears, you will find that some should actually read 'son'. The word has been TRANSLATED in some modern versions as 'child' - one might speculate as to possible reasons for this - but the underlying meaning within the New Testament when 'son' is the correct word, is that all who receive the Holy Spirit, ARE SONS. The significance is entirely that Jesus Christ was THE SON of God. If we are 'in Him', we are 'in' THE SON. He is the firstborn from the dead, the firstborn of all things, and we have been joined, through His death and resurrection, to His state of BEING the 'firstborn'. We are the 'firstborn [ones]' in Heb 12:23, therefore.
Maybe a very simple test of the difference, is in the well-known phrase 'like father, like son'. Through the Holy Spirit, we receive the 'nature' of the Son, spiritually. This is a different 'nature', spiritually, than the one Paul alludes to in Eph 2:3.
While 'children' may be an allusion to immaturity, Gal 4:1 Now I say, [That] the heir, as long as he is a child, differeth nothing from a servant, though he be lord of all; 2 But is under tutors and governors until the time appointed of the father. 3 Even so we, when we were children, were in bondage under the elements of the world: 4 But when the fulness of the time was come, God sent forth his Son, made of a woman, made under the law, 5 To redeem them that were under the law, that we might receive the adoption of sons. 6 And because ye are sons, God hath sent forth the Spirit of his Son into your hearts, crying, Abba, Father. 7 Wherefore thou art no more a servant, but a son; and if a son, then an heir of God through Christ.'
there is a deep discussion about heart-attitude to be had around the difference. This is clear in the story of the prodigal son. The elder son, who had already received his double portion, was still content to go out and work in the fields with the servants, rather than to take up his role as the heir apparent in the household. His fathe still regarded him as son, and entreated him humbly when he was angry with his brother, and the attention his brother received. The younger brother was willing to give up his 'title' as 'son', to be a servant ONLY, but the father would not hear of it.
Nevertheless, it is clear from the relationship Jesus had with His Father, into which we glimpse through John 17, and John 13, (and from the Old Testament also), that to BE a 'son', does also mean to 'obey the Father' - even as a servant of His will. Matt 7:21, John 4:34, Isa 55:9. We have a learning curve ahead. It's mainly our unrenewed minds which make it steeper than it might have been.
EDIT: When one considers the components of 'the fruit of the Spirit', they are differentiated into 'love, joy, peace, longsuffering, goodness, gentleness, meekness, temperance', so the 'children of wisdom' (as it were) may have different manifestations which the understanding can relate to wisdom, but none is 'pure wisdom' itself.
Whereas, a child of a human is a human, or, the child of a giraffe is a giraffe (for instance) - in that way identical to the parent. See also Gen 1, that 'seed' brings forth the same kind of 'tree', which bears fruit by which it can always be identified. This has a bearning on becoming like Christ, who drew attention to the difference between a seed and the results of planting it, (John 12:24), namely, the resurrection He showed us. [end EDIT].
