How do JWs respond to Jude 9 in light of Matthew 4:10?

Although I do not support JWs' non-Trinitarian doctrine, the idea that Michael could be another name for Jesus did not originate with JWs, and can be found in some writings of some early reformators.

Jude 9 should be compared to in light of Zechariah 3:2

Jude 9 (KJV):

Yet Michael the archangel, when contending with the devil he disputed about the body of Moses, durst not bring against him a railing accusation, but said, The Lord rebuke thee

Zechariah 3:1-2 (KJV):

And he shewed me Joshua the high priest standing before the angel of the Lord, and Satan standing at his right hand to resist him.

And the Lord said unto Satan, The Lord rebuke thee, O Satan; even the Lord that hath chosen Jerusalem rebuke thee: is not this a brand plucked out of the fire?

Jesus did not engage in open conversation with Satan because there is no more mercy for him. Satan's crimes will be brought against him at the judgement.

"The Lord rebuke thee" is spoken with authority. It is by His authority that Moses' body is recovered and Joshua's accuser silenced.

In Matthew 4:10, Jesus came to earth for the specific purpose of engaging with Satan - He was to withstand Satan's temptations and have victory against him. However, again Jesus did not bring against him any accusations but said "Get thee hence, Satan".

Inspiration tells us that "Death reigned from Adam to Moses" (Romans 5:14). Moses' resurrection was to be the first time that the chains of the grave are broken. So it is not inconceivable that Jesus would personally come to take His friend, whom He spoke with face-to-face, as one speaks to a friend.

Similarly in the last days, Jesus will come back Himself to retrieve us.

1 Thessalonians 4:16-17 (KJV)

For the Lord himself shall descend from heaven with a shout, with the voice of the archangel, and with the trump of God: and the dead in Christ shall rise first: Then we which are alive and remain shall be caught up together with them in the clouds, to meet the Lord in the air: and so shall we ever be with the Lord.

Both accounts in Jude 9 and Matthew 4:10 are the same person (Jesus).

Jehovah's Witnesses believe Jesus is known as Michael, The Archangel while in heaven. There are also other things to keep in mind why Jesus is believed to be the archangel.

The prefix “arch” means “principal” or “chief,” (http://www.yourdictionary.com/arch) and the word “archangel” is never used in the plural form in the Bible.

The other verse in which an archangel is mentioned is at 1 Thessalonians 4:16, where Paul describes the resurrected Jesus, saying: “16 For the Lord Himself will descend from heaven with a loud cry of summons, with the shout of an archangel, and with the blast of the trumpet of God.” (AMP Bible) So Jesus Christ himself is here identified as the archangel, or chief angel. Now, in regards to your question about Jude 9 and Matthew 4:10. These verses do not contradict.

In Jude 9, Jesus has apparently resumed his service as Michael the chief angel or Archangel. As you mentioned, Michael did not rebuke Satan in this instance. Why? Well, consider the surrounding verses before verse 9. Jude 8 says, "Nevertheless in like manner, these dreamers also corrupt the body, scorn and reject authority and government, and revile and libel and scoff at [heavenly] glories (the glorious ones)."

Apparently, there were men during Jude’s time who spoke abusively and rejected the authority of the glorious ones. Jude does not speak highly of these such men who obviously lacked humility due to haughtiness. Then, at verse 9 Jude gives a contrast of the behavior of those ungodly men and how Jesus handled a situation in regards to Moses body in a dispute with Satan. Jude was using this example to illustrate the importance of humility.

While Michael resisted Satan’s wicked scheme, he showed remarkable self-restraint. Satan surely deserved a rebuke, but Michael, who at the time he was disputing with Satan had not yet had “all the judging” committed to him, felt that such a judgment should come only from God. (John 5:22) As archangel, Michael had extensive authority. Yet, he humbly deferred to God rather than trying to seize additional authority. Besides humility, he also showed modesty, or an awareness of his limitations.

Jude was inspired to write about this incident for a reason. Sadly, some Christians in Jude’s day were not humble. They were haughtily “speaking abusively of all the things they really [did] not know.” (Jude 10) So Jude was in essence saying that if Jesus did not rebuke Satan even though he very well deserved it, why should you (these ungodly men) speak abusively of the glorious ones? Just as Jesus left it to the God to address Satan, they were encouraged to do likewise.

So, it wasn't that Michael was afraid to rebuke Satan, but rather Jesus did not want to go beyond his authority. Instead, God’s Son showed humility and modesty. He was pleased to have Jehovah God, the Supreme Judge of the universe, handle matters in His own way and time.