Computer science offers some helpful definitions:
- “Equal” does not mean “same”
- “Equal” means “same value”
Therefore, I can accurately state that (5+3) = (6+2). The expressions on each side of the equivalence are not the same, but they are equal.
The verb sometimes translated “to exercise authority over” is αὐθεντέω; it is found nowhere else in the New Testament, and could more appropriately be translated “to domineer” (see here).
The word here is ἡσυχία which connotes being tranquil & calm. It does not mean speechless—that would be σιγή (see here). It is noteworthy that the related word ἡσύχιος (again, tranquil) is applied to men & women in verse 2.
The Historical Context—Ephesus & the cult of Diana
Timothy was leading the church in Ephesus (1 Timothy 1:3), a major city in its region, where Paul himself had lived for more than 2 years (see Acts 19). This means Paul had quite a bit of personal knowledge regarding the people in Ephesus and the problems they faced.
Ephesus was home to the temple of Diana and was a hotspot for the worship of Diana (see Acts 19:23-28). In some expressions of the lore, believers in Diana saw her as the goddess of pregnancy, and would seek her protection during childbirth (see here).
The cult of Diana interpretation is interesting because Paul was just cautioning against hairstyles they were known for in verse 9 (see Stott's work here p. 84). It seems likely that this advice, and quite possibly the advice in the surrounding verses, is specifically targeted to help the church avoid domineering and immoral practices common to Ephesus and the cult of Diana.
In this case, Paul is telling Timothy, when a woman comes to you with concern about childbirth and is considering Diana worship, here is the counsel to give her: she will be saved by the true and living God, and His blessings will be poured out if she and her husband are faithful.
Note that “saved” here could mean physical protection or it could have in mind an eternal blessing, or both. In other words, whether the blessings come in this life or in eternity, they will come from the true and living God, not from Diana.
(Additional context re the cult of Diana can be found in the aforementioned work by Stott, with summaries in my thoughts on a related question here, and an article by Gregory Brown here)
The idea that there is no difference between men and women is not to be found in any ancient text that I am aware of, including the Bible. That they have the same value is found in the Bible, as noted in the OP.
The roles and identities of men & women are a major social topic today; but to expect the views of today to be found more-or-less identically-reflected in texts written millennia ago is known as the fallacy of presentism—to superimpose on the past the worldview of today.
If you’re looking for a rubber-stamp for a particular modern philosophy, a letter written circa AD 63 probably won’t give it to you. If you’re looking for profound theology, there the text may be of great value.
An educated guess on Paul's overall point - the tie in to Adam & Eve
This related question points out the unusual shifts between singular and plural in this chapter.
Verses 13 & 14 are about Adam and Eve. If they are the antecedents for verse 15, “she” is a reference to Eve and “they” means Adam & Eve. Then Paul might be saying something like this:
Redemption from the Fall will come because Eve will safely carry into the world descendants, from whom will come her Savior. Christ is of the seed of Eve, and so her Salvation is indeed a result of her motherhood (see Genesis 3:15 as well as Dottard’s post here). Her role as a mother is a critical part of God’s plan to offer salvation to her and to the entire human family.
What about the “they”? Adam isn’t off the hook here. Eve does the child-bearing, but both mother and father have a sacred duty—together—to bring up their family in “faith and love and holiness, with self-control.” The righteous branch of their posterity, from which Christ will come, is borne by Eve, but is to be raised & taught by both Adam and Eve.
Paul recognizes the vital roles played by men & women of faith. There is no "greater" or "lesser" because we "are all one in Christ Jesus," and what we have in common is greater than what we have in difference--across any set of variables--because "we are the children of God." (Romans 8:16)
For a deeper dive on how I believe God Himself feels on this issue, see my thoughts in the latter half of the video found here.