Sheol appears in the common messianic text of the Hebrew bible 65 times, but Hades only appears 11 times in the common Byzantine text type of the Greek New Testament. Although both Sheol and Hades are usually translated into the word hell, the public conception of the two places was very different in antiquity. Despite popular belief, hell is never on fire until Hades gives up its dead at the end of time and is thus thrown into the pool of fire as seen in Revelation 20:13-14.
In Genesis 37:35 Jacob refused to be comforted as a result of the evidence that suggested that Joseph was dead, and Jacob said that he would go down to Sheol in morning for Joseph. Many bibles translate Sheol in such a contexts as “the grave”, but then when people go to Sheol out of punishment often bibles will translate the word as “hell.” In Deuteronomy 32:22 we read, “For a fire is kindled in mine anger, and shall burn unto the lowest hell (Sheol), and shall consume the earth with her increase, and set on fire the foundations of the mountains.” (KJV) We thus find Sheol in Deuteronomy 32:22 to have multiple layers, and God says that he will burn the lowest layer of Sheol in his anger. Overall, the point is that even holy people like Jacob expected to go to Sheol in death, but God would burn the lowest layer of Sheol in his anger as like a act done out of jealousy as a result of Jews worshiping other gods (this would not impact Jacob if he went to Sheol).
In Numbers 16:30 Sheol is often translated as “the pit”, and this suggests that Sheol is like a place that people would suddenly fall into. In 1 Samuel 2:6 God is said to bring people both down to Sheol and up from Sheol. Job 7:9 says that people that go into Sheol will not come up from it, and 1 Kings 2:6 talks about Sheol as being a place that a person could go in peace. Overall, what is clear is that most people believed that those that go into Sheol never would return (unless God brought them out), and some would fall into Sheol unexpectedly while others would willingly go into Sheol out of peace.
Job 14:13 however talks about Sheol as being a place that God could hide people into when he is wrathful, and then later God could remember those people at a appointed time to bring them out from Sheol when God’s wrath has passed by. Job 17:13 even talks about Sheol as being a house in darkness that a person would live in, but Job 17:16 talks about Sheol having bars like a prison. Psalms 9:17 on the other hand describes Sheol as being the place that both the wicked go and all nations that forget God go into. Psalms 18:5, 30:3, 49:15, 86:13, and 88:3 describe David as being rescued from Sheol, and this gives strong evidence that God brings people out of Sheol as he desires. Overall, to some Sheol is a place that God can put a person for safe keeping to bring out later, but for others Sheol is a place that people would not like to go into and would desire to be rescued from.
Psalms 55:15 describes Sheol as being a dwelling of wickedness, and David asks God to make the wicked dwell in Sheol. Psalms 139:8 however describes God as being both in heaven and in Sheol, and this also follows along with the message of Proverbs 15:11 to say that Sheol and destruction are before the Lord. Proverbs 7:27 & 9:18 describe Sheol as being a place that people who engage in sinful lifestyles go into, and Proverbs 27:20 & 30:16 both talk about Sheol as being a place that is never filled as if it constantly consumes without being satisfied. Overall, despite the fact that God is seen both in heaven and in Sheol and despite the fact that Sheol is a place for the wicked, Ecclesiastes 9:10 talks about everyone going to Sheol at some time in their life as it describes Sheol as being a place with no work, no things inside it, no knowledge, and no wisdom.
Song of Songs 8:6 talks about Sheol as being a wicked jealous place, and the request by the bridegroom to the bride that is given to prevent the bride from descending into Sheol is that she seal a covenant on her heart and on her arm. Just as the New Jerusalem is the bride of the lamb, thus the Church as the New Jerusalem can be prevented from falling into Sheol through being in covenant relation to the lamb (Christ). In fact Isaiah 28:15 talks about people making a covenant with both death and Sheol and people are considered to be making lies their refuge, and thus we find Isaiah 28:15 to be in stark contrast to Song of Songs 8:6. Overall, a covenant thus exists that protects people from Sheol, but then also a false covenant exists that is a false protection from Sheol.
Isaiah 28:18 talks about God canceling the covenants that people had with Sheol, and Isaiah 38:18 goes so far as to say that people in Sheol cannot give praise or even hope for the truth of God. Ezekiel 31:15-17 talks about God casting many into Sheol from Eden (the garden of God), and Ezekiel continues to talk about the uncircumcised with their weapons of war going down to Sheol as seen in Ezekiel 32:21 & 32:27. Hosea 13:14, Amos 9:2, and Jonah 2:2 all describe God ransoming people out of Sheol, but what is unique about Amos 9:2 is that even though people dig themselves into Sheol God will take them out just as he pulls down those that try to climb to heaven. Overall, the point to be made is that God is completely in control in putting people into Sheol and talking people out of Sheol, and this work of God happens despite the best efforts that people make.
Hades is different from Sheol in that instead of being just a simple holding place of souls instead in ancient Greek mythology it has multiple divisions such as Elysium (isles of the blessed), Asphodel Meadows (land of neutrality), Tartarus (the dungeons), and Erebus (pure darkness). Although I am not a accomplished scholar of Greek mythology, I will point out that many Greeks most likely believed Hades to be the afterlife that comprised what Christians find to be both heaven and hell. Jesus Christ however makes clear that Hades is the place that unrepentant people go as seen in Matthew 11:23, and Luke 10:14-15 that talks about Hades as being almost unbearable punishment. Overall, the way that Jesus Christ redefines Hades in the New Testament shows a strong condemnation associated with Hades, and this is especially true as seen in the story of the rich man in Hades being tormented by the fire as seen in Luke 16:23.
Matthew 16:18 shows Christ bestowing the foundation of the church on Peter as he says that the gates of Hades will never overcome the church, and this shows us that unlike Greek mythology Christ does not understand Hades to contain heaven. Acts 2:25-31 in a way shows how David placed his hope in Christ and thus he was not left in Hades but instead was risen out of it, and the fact that this scripture refers back to the Hebrew scripture when it uses Sheol tells us that the authors of the Greek New Testament saw Hades and Sheol as being rather interchangeable in speech. Overall, culturally both Hades and Sheol had been very different because Hades in Greek mythology was named after a ruler named Hades that ruled over the underworld, and Hades thus was much more of a personalized place with both eternal rewards and punishments.
1 Corinthians 15:54-55 shows us that Hades no longer has victory, and the conquering of Hades is seen in Revelation 1:18 as Christ describes himself as having the keys to Hades. In Revelation 6:8 however we see Hades as actively going to war as like a person that tries to conquer, and both in Revelation 1:18 & Revelation 6:8 we see death as described as like a person that goes along with Hades. Overall, we only see Hades being cast into the pool of fire at the very end of time when everything gives up its dead as seen in Revelation 20:13-14.
I find that the reason why many bible translators will translate Sheol as the grave or as the pit in some places and hell in other places is because they would like to make the distinction between people in the Hebrew scripture receiving rewards and those receiving punishments, but the problem with doing this is that people get the conception that good patriarchs went to one place while wicked people went to another place in the Hebrew scripture. Even though the Greek conception of Hades could be similar to that of Sheol, Christ described something far better than Sheol and in doing so he spoke harshly of Hades. Revelation 21:1-2 talks about the old system of heaven and hell passing away as God makes a new heaven and a new earth, and this demonstrates why Christ would be so admit about condemning and destroying those that would otherwise be in Hades in the second death (pool of fire) as described in Revelation 21:8. Overall, the point is that hell has served many purposes throughout the scripture, but in the end both heaven and hell are completely rearranged (or even removed) and that is why the New Testament and the Old Testament talk about hell in very different ways.
