What's the biblical basis for a difference between children and offspring of God, if any?

The idea that only Christians are children of God does not hold up. While I would admit one can interpret the Bible to say that, the preponderance of the evidence is against it.

For Christians, Jesus was the Son of God. Was he a Christian? Obviously not because he never needed salvation as Christians admit they do. This, of course is a special case, so let's look at other examples.

In the OT, the term "sons of God" is used frequently to refer spiritual beings and perhaps people who are not Christians.

Deuteronomy 32:8 When the Most High gave to the nations their inheritance, when he separated the sons of men, he fixed the bounds of the peoples according to the number of the sons of God.

In the Book of Job, the term "sons of God" definitely refers to beings we usually think of as angels.

Now there was a day when the sons of God came to present themselves before the Lord, and Satan also came among them. (Job 1:6)

In Exodus 4:22-23 and elsewhere, God calls the Israelites collectively "My son."

And you shall say to Pharaoh, ‘Thus says the Lord, Israel is my first-born son, and I say to you, “Let my son go that he may serve me”...

Several verses show God referring to Solomon as "my son."

He shall build a house for me, and I will establish his throne for ever. I will be his father, and he shall be my son; I will not take my steadfast love from him, as I took it from him who was before you, but I will confirm him in my house and in my kingdom for ever and his throne shall be established for ever.’” (1 Chronicles 17:12-14)

The New Testament calls Adam "the son of God."

Luke 3:38 - ...the son of Enos, the son of Seth, the son of Adam, the son of God.

Finally in Matthew 5:9 Jesus says: “Blessed are the peacemakers, for they shall be called sons of God." It does not appear in context that Jesus was teaching that only Christians can be peacemakers.

Conclusion: although @Alex Strasser has shown that there is an interpretive Biblical basis to believe that Christians alone are "children of God," there is direct Biblical evidence that the term also applies to Jesus, angels, Adam, the Israelites, Solomon and peacemakers generally.**

Is every human a child of God? Personally I would say yes. God loves all mankind as a Parent, and even "lost" souls belong to Him. In Ezekiel 18:4 God says, "Behold, all souls are mine; the soul of the father [who sins] as well as the soul of the son [who does not] is mine."

Summary

Everyone on Earth are the "offspring (descendants) of God," but only Christians are the "children of God" or "sons of God."

Offspring of God

The primary basis for the teaching that everyone on Earth are offspring (descendants) of God is in Paul's address to the Gentile Areopagus, which was full of non-Christians.

Acts 17:28-29 (ESV) 26 And he made from one man every nation of mankind to live on all the face of the earth, having determined allotted periods and the boundaries of their dwelling place, 27 that they should seek God, and perhaps feel their way toward him and find him. Yet he is actually not far from each one of us, 28 for “‘In Him we live and move and have our being’; as even some of your own poets have said, “‘For we are indeed his offspring.’ 29 Being then God's offspring, we ought not to think that the divine being is like gold or silver or stone, an image formed by the art and imagination of man."

The Greek word for offspring is γένος (genos, Strong’s 1085). This word is never translated as either "children" or "sons." The NASB breakdown of translation from the link above is

birth (2), countrymen (2), descendant (1), descent (1), family (2), kind (3), kinds (3), nation (1), native (1), race (3).

The takeaway is that since God created Adam and we all descended from Adam, we are all descended from God as perhaps the "Father" of all humanity in that sense. It carries with it little connotation of being adopted into God's family, which is unique to God's people, His bride, the Church aka Christians.

Children of God

The phrase “children of God” appears 9 times in Scripture (ESV). One clear verse to explain to whom this phrase applies is John 1:12

John 1:12 (ESV) But to all who did receive him, who believed in his name, he gave the right to become children of God.

The explicit separation of the Christians from non-Christians is (not a ego-trip) based in verses such as Romans 9:8

Romans 9:8 (ESV) This means that it is not the children of the flesh who are the children of God, but the children of the promise are counted as offspring.

and 1 John 3:10 (Note that Romans 9:8 ends in the word “offspring”, which is a completely different Greek word, σπέρμα (sperma, Strong’s 4690)

1 John 3:10 (ESV) By this it is evident who are the children of God, and who are the children of the devil: whoever does not practice righteousness is not of God, nor is the one who does not love his brother. The Greek word for children is τέκνον (teknon, Strong’s 5043). The word is used 90 times, and it is translated as “children” or “sons.” It is never translated as descendants or offspring. From Thayer’s Greek lexicon (see linked Strong’s), some definitions of this word help convey the theological concepts and implications of the phrase, such as:

1. with emphasis: to be regarded as true, genuine children; children begotten by virtue of God’s promise 2. the name is transferred to that intimate and reciprocal relationship formed between men by the bonds of love, friendship, trust, just as between parents and children; in affectionate address; pupils or disciples are called children of their teachers, because the latter by their instruction nourish the minds of their pupils and mold their characters; those to whom, as dearly beloved of God, he has appointed salvation by Christ; those whom God knows to be qualified to obtain the nature and dignity of his children

It is not about Christians vs non-Christians, but God has a unique relationship and a unique love for His Church, whom is His Bride. The emphasis on the “children of God” is on the promise, inheritance, unique love, and unique relationship with those whom are submissive to the true living God and reconciled through His Son Jesus Christ. Jesus is the method of reconciliation of all people to Himself, and He made that message open and available to all.

Sons of God

The most important thing to point first and foremost is that no mere human is ever called “the Son of God.” Jesus Christ is the only whom to whom the title is given “the Son of God” (ὁ Υἱὸς τοῦ Θεοῦ) such as in Mark 3:11. Colwell’s Rule also applies here. The “the” is important. The Church has its members called “sons of God.”

The word for son in Greek is υἱοῖς (huiois, Strong’s 5207). From the HELPS word studies (for what it’s worth), the accompanying sentence is that “for the believer, becoming a son of God begins with being reborn (adopted) by the heavenly Father.” The NASB breakdown for translation is

attendants (3), foal (1), man (1), son (1), son (307), sons (68).

Speaking to the church in Galatia, Paul says in Galatians 3:26

Galatians 3:26 (ESV) For in Christ Jesus you are all sons of God, through faith.

And Paul to the church in Rome in Romans 8:14

Romans 8:14 (ESV) For all who are led by the Spirit of God are sons of God.

Jesus is exceedingly set apart as the Son of God compared to us. We are not the Word that was with God in the beginning and was God (John 1:1) who became flesh and dwelt among us (John 1:14), through whom all things were created and for whom all things were created (see Colossians 1:15-23). He is the one whom is given the title of God (Theos) 7 times in Scripture. See my other answer on Jesus claims and titles of Son of God and God here.

The topic of us being “sons of God” invokes the notion of adoption (Ephesians 1:5, read an article by John Piper. We pray to Jesus because He is God. We only only only pray to God (protestant view). “We only have one mediator between God and man, the man Christ Jesus” (1 Timothy 2:5). I hope the above helps you see that Jesus is the unique Son of God such that we in no way are Jesus’ brothers even in spirit.

Titles for the Unbeliever

While the lost should not and cannot accurately be given the titles “children of God” or “sons of God,” the Bible gives them several titles invoking the same words “children” and “sons.” Many of these titles come from Ephesians 2:1-10, which may possibly (in my opinion) be the best summary of the Gospel, the main message of God to humanity, in all the Scriptures.

Ephesians 2:1-10 (ESV) And you were dead in the trespasses and sins 2 in which you once walked, following the course of this world, following the prince of the power of the air, the spirit that is now at work in the sons of disobedience— 3 among whom we all once lived in the passions of our flesh, carrying out the desires of the body and the mind, and were by nature children of wrath, like the rest of mankind. 4 But God, being rich in mercy, because of the great love with which he loved us, 5 even when we were dead in our trespasses, made us alive together with Christ—by grace you have been saved— 6 and raised us up with him and seated us with him in the heavenly places in Christ Jesus, 7 so that in the coming ages he might show the immeasurable riches of his grace in kindness toward us in Christ Jesus. 8 For by grace you have been saved through faith. And this is not your own doing; it is the gift of God, 9 not a result of works, so that no one may boast. 10 For we are his workmanship, created in Christ Jesus for good works, which God prepared beforehand, that we should walk in them.

Another relevant verse, which is also mentioned above, is 1 John 3:10

1 John 3:10 (ESV) By this it is evident who are the children of God, and who are the children of the devil: whoever does not practice righteousness is not of God, nor is the one who does not love his brother.

And Romans 5:10

Romans 5:10 For if, while we were God’s enemies, we were reconciled to him through the death of his Son, how much more, having been reconciled, shall we be saved through his life!

Some of the titles given to unbelievers in the Scripture are:

1. Children of wrath (Ephesians 2:3)
2. Children of the devil (1 John 3:10)
3. Sons of disobedience (Ephesians 2:2)
4. Offspring of God (Acts 17:28-29)
5. Dead in sins (Ephesians 2:1, 5)
6. Lost (Luke 19:10)
7. Enemies of God (Romans 5:10)

I think the point of appreciating this list is both in retrospective and prospective: being overwhelmingly thankful and grateful to God from rescuing us from our past as (insert one of the titles), when we weren’t saved, and also prospective to how that should affect our earnestness and eagerness to share the Gospel such that others will know this message and become adopted into God’s family. Also not that though unbeliever’s are called God’s enemies and children of wrath, God demonstrated His love for us/them while we were still sinners (Romans 5:8), so He still loves everyone on the face of the earth.

I didn't read Alex's entire post, but what I did read is accurate.

From a logical standpoint it boils down to 2 things and there is tension there. Biological representation and Influential representation.

Someone that is biologically my offspring, but that I do not raise, will not be like me in their mannerisms and expressions and way of life. Nature and nurture both matter.

While all humans are physically made in God's image, not all humans represent God the way Jesus does because Jesus has spent time with The Father in a way that they have not. Anybody can gain access to the Father and His influence through Christ. People that are Christians are people that have been born again, and have started down the path of being influenced by God beyond having their biology created in His image.