Would a cessationist and continuationist pray any differently for someone to be healed?
It may be helpful to first identify what is a cessationist? If you believe the miraculous spiritual gifts that the Lord granted to the apostles to validate the start of the church, including the ability to speak the gospel and have it heard and understood in various languages (a reversal of the curse at Babel) were operative in the apostolic era only ... and that some and eventually all of those particular gifts gradually ceased prior to the end of the first century, then you are a cessationist.
If you believe the miraculous gifts of the Spirit described in the New Testament (i.e. Ephesians 4) have continued unabated, unchanged, and unaltered since the initial outpouring of tongues at Pentecost, you are a continuationist.
Let me say at the outset, many who identify themselves as continuationists are actually partial cessationists. In other words, they do not declare one another “apostles” or claim all the apostolic prerogatives as some do. For example, they do not claim to be able raise people from the dead and they do not twist and/or corrupt nearly every category of doctrine related to the gospel, from the atonement to Christian discipleship, as some do.
All evangelicals believe that the canon of Scripture is closed, right? We do not believe we should be seeking to add new inspired material to the New Testament canon, right? We hold to the faith that was once for all delivered to the saints (Jude 3)—in the person of Christ, and through the teaching of His apostles, and has been inscripturated in the New Testament. In other words the canon is complete. Those who do not believe that are not really evangelicals. They are either cultists, heretics or false teachers who seek to add to the Word of God.
But notice this: if you acknowledge that the canon is closed and the gift of apostleship has ceased, you have already conceded the heart of the cessationist's understanding of scripture. And some leading “Reformed continuationists” go so far as to admit that all the charismatic gifts in operation today are of a lesser quality than the gifts we read about in the New Testament.
For example, in Wayne Grudem’s book The Gift of Prophecy in the New Testament and Today (Wheaton: Crossway, 1988), which may be the single most influential work written to defend modern prophecy, Grudem writes: “no responsible charismatic holds” the view that prophecy today is infallible and an inerrant revelation from God (p. 111). He says charismatics argue for a “lesser kind of prophecy” (112), which is not on the same level as the inspired prophecies of the Old Testament prophets or the New Testament apostles which was infallible. Grudem claims, there is almost uniform testimony from all sections of the charismatic movement that [today's] prophecy is impure, and will contain elements which are not to be obeyed or trusted.
Jack Deere, former Dallas Seminary professor turned charismatic advocate, admits in his book Surprised by the Power of the Holy Spirit (Grand Rapids: Zondervan, 1993), that he has not seen anyone today performing miracles or possessing gifts of the same quality as the signs and wonders of the apostolic era.
Deere and Grudem have, in effect, conceded the entire cessationist argument. They have admitted that they are themselves cessationists of sorts. They believe that the true apostolic gifts and miracles have ceased, and they are admitting that what they are claiming today is not the same as the charismata described in the New Testament. Contemporary tongues-speakers do not speak in understandable or translatable dialects, the way the apostles and their followers did at Pentecost. Charismatics who minister on the foreign mission-field are not able to preach the gospel miraculously in the tongues of their hearers. They have to go to language school like everyone else.
In Thomas Edgar's Satisfied by the Promise of the Spirit (Grand Rapids: Kregel, 1996), he writes: The charismatic movement gained credence and initial acceptance by claiming their gifts were the same as those in Acts. For most people this is why they are credible today. Yet now one of their primary defenses, is the claim that [the gifts] are not the same [as those in the New Testament.] Faced with the facts, they have had to revoke the very foundation of their original reason for existence. (p. 32)
"The signs of a true apostle were performed among you with utmost patience, with signs and wonders and mighty works." 2 Corinthians 12:12 (ESV)
Scripture says the miracles were apostolic signs; and therefore, by definition they pertained specifically and uniquely to the apostolic era.
So to answer your question, how would these two groups pray differently? True continuationists would claim promises the Bible does not grant to them, commanding things in Jesus name that exceed their biblical authority, while a cessationist would boldly but humbly pray for God's will to be done, believing, knowing and trusting with absolutely confidence that the Lord is able and capable of healing anyone He so chooses; but whatever the outcome, it will be for our good and His glory.
(Thanks to Phil Johnson for some of the material used above)