Note that the Greek word for "falling away" in 2 Thessalonians 2:3 is apostasia; its Greek definition is "defection, revolt, especially in religious sense, rebellion against God, apostasy."
In 2 Thessalonians 2:1-8, I believe Paul makes clear that the apostasy and abomination of desolation must occur before the day Jesus comes to gather us together, for Jesus' coming (parousia) to gather us together must destroy the Antichrist (verse 8). I believe Paul is referring to the same coming and gathering together as Matthew 24:29-31. I don't believe Paul taught a 3rd coming or a 2nd rapture. We Christians must go through the reign of the Antichrist (Revelation 13:7-10, 14:12-13).
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Departure
Note that apostasy is a departure, from one's faith.
"The Spirit speaketh expressly, that in the latter times some shall depart from the faith" (1 Timothy 4:1); "Then shall many be offended, and shall betray one another, and shall hate one another. And many false prophets shall rise, and shall deceive many. And because iniquity shall abound, the love of many shall wax cold" (Matthew 24:10-12); compare "From that time many of his disciples went back, and walked no more with him" (John [b]6:66[/b]).
Some say apostasia can also refer to "a physical departure." But note that apostasia (2 Thessalonians 2:3) isn't used in the Bible or in other ancient Greek literature to refer to people departing from a physical location, but is used in the ancient Greek translation of the Old Testament (for example, Joshua 22:22, 2 Chronicles 29:19, Jeremiah 2:19) to refer to people departing from their religion, and is used only one other time in the New Testament: Acts 21:21, where it's translated "forsake." Look at its Greek definition and you will see why: "defection, revolt, especially in religious sense, rebellion against God, apostasy."
Paul was trying to clear things up for the Thessalonians, not confuse them by using a word in a way that it had never been used before and would never be used again.
Note that even Darby himself acknowledged that apostasia means apostasy: "Let not any one deceive you in any manner, because [it will not be] unless the apostasy have first come, and the man of sin have been revealed, the son of perdition" (2 Thessalonians 2:3, Darby Translation).
Note that aphistemi and apostasia aren't interchangeable; they're not simply the verb and noun form of the same word. Rather, they're two different words with two different definitions: apostasia is a "defection, revolt, especially in religious sense, rebellion against God, apostasy" whereas aphistemi is to "put away, remove, keep out of the way."
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Shaken Thessalonians
I believe that in Paul's day some false teachers were causing the Thessalonians to be "shaken in mind, or be troubled" (2 Thessalonians 2:2) by spiritualizing away the 2nd coming, just as full preterists do today, and just as those who spiritualized away the resurrection in Paul's day were destroying the faith of some (2 Timothy 2:18).
Some say the Thessalonians were shaken about possibly missing the rapture and being in the tribulation. But the Thessalonians wouldn't be "shaken in mind, or be troubled" (2 Thessalonians 2:2) about being in persecutions and tribulation, for they were already patiently enduring both: "We ourselves glory in you in the churches of God for your patience and faith in all your persecutions and tribulations that ye endure" (2 Thessalonians 1:4), but they would be shaken if the 2nd coming had supposedly already happened, because where was Jesus? And if the 2nd coming had already happened, then the resurrection must have already happened as well, and where were all their departed loved ones that Paul had promised them they would see again (1 Thessalonians 4:13-18)? Those who spiritualized away the resurrection in Paul's day were destroying the faith of some (2 Timothy 2:18).
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"The" Falling Away
I believe that Paul mentions "the" falling away for the same reason as he mentions "the" man of sin, that he'd told the Thessalonians of both of these things before in person (2 Thessalonians 2:5).
In 2 Thessalonians 2, Paul doesn't promise to speak of "our gathering together unto him" separately from "the coming of our Lord Jesus Christ" (verse 1), but refers subsequently to both as "the day of Christ" (verse 2) and "that day" (verse 3).
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Apostasy
I believe apostasy is when someone who was saved departs from their faith, and so loses their salvation:
"Thou standest by faith. Be not highminded, but fear: For if God spared not the natural branches, take heed lest he also spare not thee. Behold therefore the goodness and severity of God: on them which fell, severity; but toward thee, goodness, if thou continue in his goodness: otherwise thou also shalt be cut off" (Romans 11:20-22).
"Take heed, brethren, lest there be in any of you an evil heart of unbelief, in departing from the living God. But exhort one another daily, while it is called To day; lest any of you be hardened through the deceitfulness of sin. For we are made partakers of Christ, if we hold the beginning of our confidence stedfast unto the end" (Hebrews 3:12-14).
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Can Christians Apostatize?
Some say a true believer will never depart from the faith. Then how do we know we're a true believer unless we endure to the end? (Matthew 24:13)
"The Spirit speaketh expressly, that in the latter times some shall depart from the faith" (1 Timothy 4:1), and this departure from the faith (the falling away or apostasy) will happen before Jesus comes to rapture us (2 Thessalonians 2:1-3). How can someone apostatize from a faith they never had?
Some say apostasy isn't possible for Spirit-filled Christians. But there are no Christians without the Spirit (Romans 8:9), and even Spirit-filled Christians can commit apostasy and unrepentant sin (Hebrews 6:4-6, 10:26-30). The Holy Spirit doesn't turn us into robots.
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The Abomination Of Desolation
I believe "exalteth himself above all that is called God, or that is worshipped; so that he as God sitteth in the temple of God, shewing himself that he is God" (2 Thessalonians 2:4) refers to when "they shall place the abomination that maketh desolate... and he shall exalt himself, and magnify himself above every god" (Daniel 11:31, 36), and so refers to "the abomination of desolation, spoken of by Daniel the prophet" (Matthew 24:15).
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Treason
Imagine that a prophecy said "treason must come first." Would some say that could mean God will first "hand us over" from the earth into heaven because "treason" is derived from the same root word as "to hand over"?
Note that two words can be derived from the same root and still have different definitions: "Treason: 12th century. Via Anglo-Norman treisoun , 'treacherous handing over, betrayal,' from the Latin stem tradition, from tradere, 'to hand over' (source of English traitor, betrayal, and tradition)."
I believe that someone saying "apostasia" in 2 Thessalonians 2:3 can mean the same as "aphistemi" because they're derived from the same root word is like someone saying: "While the prophecy says 'treason must come first,' note that 'treason' is derived from the same root word as 'to hand over,' which doesn't have to be bad since 'tradition' is derived from 'to hand over' because we 'hand over' a tradition from one generation to the next. So I feel I am justified in saying that the prophecy could really be saying 'a handing over must come first,' that is, God will first 'hand us over' from the earth into heaven."
Wouldn't someone be justified in asking why the prophecy would have used "treason" to refer to a good handing over? Similarly, wouldn't someone be justified in asking why the prophecy would have used "apostasia" to refer to a good departure, instead of a departure from the faith?
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Discessio
Note the definition of the Latin (Vulgate) word discessio: "In the church, a separation, schism (eccl. Lat.), Vulg. Act. 21, 21; id. 2 Thes. 2, 3."
(from http://www.perseus.tufts.edu/cgi-bin/ptext?doc=Perseus%3Atext%3A1999.04.0059%3Aentry%3D%2314158 )
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The Day Of Christ
Some say the day of the Lord isn't the day of Christ (2 Thessalonians 2:2). But I believe they're the same, the day of the Lord Jesus Christ (1 Corinthians 1:7-8), the 2nd coming, the day he comes to gather us together (2 Thessalonians 2:1, Matthew 24:31).