I see three main differences in the two expositions. From the first to the third, the differences grow stronger because the implications build on top of themselves

SECTION ONE: Hodge on verse 2.

If then, asks the apostle, such a destiny as this awaits you, are ye unfit to decide the smallest matters?

If the world (mankind) shall be judged by you (ejnuJmi~n) , i.e. before you as judges. Are ye unworthy (ejnuJmi~n), i.e. of too little weight or value, having neither the requisite dignity nor ability. Unworthy of the smallest matters. The word (krith>rion), here rendered matters, in the sense of causes, or matters for judgment, means,
1. A criterion or test; a rule of judgment.
2. A tribunal or place of judgment, and then, the court or assembled judges. Exodus 21:6. Judges 5:10. Daniel 7:10, and in the New Testament, James 2:6.
3. The trial, i.e. the process of judgment.
4. The cause itself, or matters to be tried.

This last sense is doubtful, although it is generally adopted here because it suits so well the fourth verse, where the same word occurs. The second sense would suit this verse. ‘If ye are to sit with Christ on the seat of universal judgment, are ye unworthy of the lowest judgment seats.’ But the fourth verse is in favor of the explanation adopted in our version. ‘Are ye unfit for the least causes?’

Hering on Verse 2:

The shameful conduct is defined, first of all (and in truth further aggravated), by the fact that these disputes are not even serious in nature, but trivial. The Apostle asks them in 2b, as future judges of the cosmos, “Are you not capable of [judging] the most trivial cases?” The adjective found here is the superlative form of mikrÒj, meaning “the smallest”. Paul expects the believers in Corinth to settle such embarrassingly small squabbles outside of the civil courts.

Summary: Hodge renders verse 2, “’If ye are to sit with Christ on the seat of universal judgment, are ye unworthy of the lowest judgment seats.’” whereas Hering would render the end, “are you not capable of judging the most trivial/smallest cases.” Hering assumes the interpretation of the word (krith>rion) to be ‘judging’.

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SECTION TWO: Hodge on verse 4: Christians should act in reference to them in a manner consistent with their high destiny.

Set them to judge who are least esteemed in the church. The original admits of this translation. If the passage be so rendered, then it has a sarcastic tone. ‘Set your least esteemed members to decide such matters.’

It may, however, be read interrogatively, ‘Do ye set as judges those least esteemed in (i.e. by) the church (that is, the heathen)?’ This translation is generally preferred as best in keeping with the context. The sentence is emphatic. ‘Those despised (see 1:28) by the church, – those do you set to judge?’ It is an expression of surprise at their acting so unworthily of their high calling.

Hering on verse 4: Arguing from the greater to the lesser, he appeals to their sense of shame by accentuating the dissonance found between their future cosmic role and their present behavior. Important to note in this first section of our passage is that he wishes to impress upon the hearers the unessential nature of the litigation. He repeats, and thereby underscores, this characteristic in verses 3 and 4. The litigants’ concerns are described twice as biwtik£ and biwtikὰ…krit»ria, “everyday matters of this life.”

Paul’s directive to the church is clear: The church should not be taking common and trivial matters before the magistrate. The shame, then, is not in the litigation itself (the grievances, though small, are not imagined).4 The focus of Paul’s criticism is that they have not found among themselves the capacity to try even trivial matters. The civil cases are the secondary, undesirable outcome of this central malady.

If this is indeed the case, then it would follow that Paul might allow a Christian to be engaged in litigation involving a more significant grievance, which exhausts the capacity of the church to adjudicate, or falls outside of its juridical purview.

Summary: Hodge shows that those in the Church due to their high calling (v.2 – to judge angels) should deal with all litigation matters, and at the same time argues that Christians shouldn’t put litigation before Civil Magistrates who are least esteemed in the Church. Hering focuses on taking litigation which is trivial before the Civil Magistrate and concludes that ‘trivial’ matters shouldn’t be taken. Therefore, Paul might allow a Christian to be engaged in litigation before a Civil Magistrate.

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SECTION THREE: Hodge on verse 7: 7. Now therefore there is utterly a fault among you, because ye go to law one with another. Why do ye not rather take wrong? Why do ye not rather (suffer yourselves to) be defrauded?

Now therefore [h]dh me, already indeed therefore. That is, these lawsuits are already, or in themselves (o[lwv), an evil irrespective of their being conducted before heathen judges. The word (h[tthma]]) does not so properly mean fault as loss or evil. It is a loss or evil to you to have these litigations. See Romans 11:12, where the rejection of the Jews is called their (h[tthma) loss.

Why do you not, etc. That is, why, instead of going to law with your brethren, do you not rather submit to injustice and robbery? This is a clear intimation that, under the circumstances in which the Corinthians were placed, it was wrong to go to law, even to protect themselves from injury. That this is not to be regarded as a general rule of Christian conduct is plain, because, under the old dispensation, God appointed judges for the administration of justice; and because Paul himself did not hesitate to appeal to Cesar to protect himself from the injustice of his countrymen.

Hering on the same issue: Likewise, if a matter of dispute cannot be decided in a church court (including the process of mediation, where applicable) due to an intractable division of opinion, then the church has no choice but to seek out the civil courts if an equitable conclusion is to be achieved.

These are generally serious matters, not to be confused with the everyday tussles treated in 1 Cor. 6:1-6. The court case would then be brought from the injured church member against the church or its members…An appeal in such cases to the peace of the church, or, as in our passage, v. 7, “to overlook” these sins, makes light of the matter of sin and its consequences. When the church abdicates her responsibilities, coddles sin or condones unrighteousness, her violated members must turn elsewhere for truth and justice. But is this allowed by Paul? The Apostle turns from the trivial matters of verse 1-6 and simultaneously raises the stakes by asking why the believers do not simply “allow themselves to be treated with injustice (ἀdikeῖqe)?” “Why not, yet the more, to be defrauded (ἀpostrereῖsqe)?” This manner of argumentation, so familiar in Paul, assumes that the previously mentioned trivial cases, fraught with selfish concerns, are to wither in significance in light of the suffering of injustice (and all that this might imply). In contrast to their petty disputes, Paul introduces the weighty matter of genuine suffering and significant loss. He poses a sincere question, which certainly opens the possibility for the believer “to overlook” (and we assume forgive) even such grievous personal harm as the words imply. Note that here, as with the rest of our passage, he simply poses a question to the suffering believer; it is not, as popularly assumed, a command from the Apostle. There is no indication that this question should be the foundation for dismissal of all inter-Christian grievances, or a prohibition against all civil litigation.

The question is posited, as with other rhetorical questions in Paul, to engender appropriate attitudes and behaviors, not to eliminate the possibility of righteous litigation. In spite of this it remains a legitimate goal for a Christian to appeal to this ethic in the face of injustice or wrongs committed. It is a praiseworthy Christian ideal, often executed with the highest of virtue and at great personal cost. It is illegitimate, however, for believers to call upon this text to embolden acts which are unjust or result in others’ loss. The verse is directed solely to those who have been treated with injustice or have been defrauded. An exegesis which allows appropriation of Paul’s ethic for personal gain is nothing more than presumption and arrogant self-indulgence, the very things against which the Apostle is reasoning. Indeed, we are sternly warned that such behavior involves more than personal loss; it invites divine retribution [v.9].11

Summary: Hodge shows that taking lawsuits between brothers before civil magistrates is an evil. Rather, we should suffer injustice/robbery before ever going to the civil magistrate. In other words, he is emphasizing the need to handle matters between brothers in the Church because of verse 2. The examples of Paul going to the Civil Magistrate is within the general rule because it was not ‘brother against brother.’ What Hodge is truly saying is not that we have to suffer for peace, but that Christians should be able to handle any case between brothers because of v.2. Hering, however, believes that in serious matters a brother has no choice but to take it to the civil magistrate. And, seemingly attacking Hodge’s point, to appeal for peace in the Church overlooks sin. Even if it is a praiseworthy Christian ideal to suffer for the peace of the Church, for Hering it is illegitimate.