For dispensational Baptists, the church is a sidebar in God's redemptive intent, thus God doesn't have to operate the same way in the church as he did in the past to Israel. This can either lead one to posit that God is doing a new thing within the church and thus the Spirit is operating even more supernaturally than in the pre-Apostolic age, or one might claim (as some of the Macarthurites do) that God doesn't work supernaturally hardly at all. For either group, however, there's a extreme difference in God's working in the church era versus prior.
For so-called Reformed Baptists, those who recognize the covenantal structure of God's redemptive plan but don't see the unity within the covenantal structure, there exists also a substantial difference in how God operates toward his covenant people in the church era versus how he dealt with the patriarchs. Thus, Credobaptist cessationism (and/or continuation) therefore primarily rests in a substantive difference in the way God communicates to his covenant people and in the operation of his Holy Spirit.
While I believe Baptistic cessationism rightly notes that though God communicated to his people through prophets in visions and dreams until the end of the Apostolic era, to suggest that this expresses an end of the prophetic gift altogether is perhaps misguided.
Firstly, I want to note that the terms continuationism and cessationism are problematic when speaking from a covenantal perspective. Covenantally, the Holy Spirit has been operational in redeeming, regenerating, and preserving the elect from the beginning. The primary difference brought about by the Holy Spirit's fall at Pentecost was not one of substance but rather of intent. The Spirit empowered the believers at Pentecost for the reformation and governing of God's covenant people as the church, it also caused them to fulfill Isa 29:1-13, bringing judgment against unbelieving Israel. Thus tongues was a specifically time-limited gift, intended to warn Israel of judgment (1 Cor 14:20-22), and prophecy fore-telling nature was also time-limited as the nature of the Gospel was still being revealed to the Apostles.
Thus when we read the Romans 12 list of Spiritual gifts, we see a list of God's gracious gifting to his covenant people for the organization and governing of the church. These same gifts operated within the covenant community in the Old Testament, though not always as visibly and continually as today. We still see prophecy listed therein, and while I acknowledge that the gift itself might be time limited (*see below), I tend to view it as referring to speaking God's Word to his covenant people. Something that should be happening every time a teaching elder enters the pulpit to preach.
Thus the Reformed view of the Spirit's work within his covenant people is not one primarily of change, or even cessation, rather of continuation and strengthening. God speaks today to his people through his gifted messengers, yet that message is always of Christ and his redemptive work on our behalf. This is part of Paul's often mentioned "mystery" the revealing of the Gospel and the unified covenant people of God.
We therefore need a different perspective on the nature of spiritual gifts and the work of the Holy Spirit within the church. While correction is certainly needed toward those 'continuationists' who cannot consistently maintain the doctrine of Sola Scriptura, and who misunderstand the nature and purpose of tongues, I do not believe it can be rightly heard from those who likewise maintain a substantive difference in God's covenant application.
Consider for a moment that the aforementioned gift-list found in Rom. 12 is current, and that each believer in the church has been gifted in some manner, as therein described, by the Spirit to accomplish the intended work of God in the church. This means that when a pastor of a congregation speaks to God's people on a Sunday morning, he's doing so by the power and means of the Holy Spirit who has gifted him to speak God's Word to His people, perhaps prophetically (*see below). The Sunday school teacher is empowered by God to explain the Scriptures to his students. The person gifted with helps is able to compassionately help those in need, etc. These are supernatural gifts even though they may appear as ordinary to us.
Thus the Reformed are truly continuationists, however, Biblical continuationists who recognize God's intent in spiritual gifts.
If we view the church as a completely new body, unconnected covenantally to the former people of God, or if we view the church as tangibly different in nature than the Old Testament people of God (ie: made up of regenerate people only), then we will have a skewed understanding of the Holy Spirit and his work therein.
* Note: I recognize that some argue for the end of the gift of prophecy per Hebrews 1, and I am willing to admit that possibility, however I tend to view prophecy as Calvin did:
"Hence prophecy at this day in the Christian Church is hardly anything else than the right understanding of the Scripture, and the peculiar faculty of explaining it, inasmuch as all the ancient prophecies and all the oracles of God have been completed in Christ and in his gospel."Calvin's understanding of Romans 12, in fact, is an excellent argument for the supernatural nature of the gifts working in the church today. Calvin views the gifts in different categories as they apply to the governing of the church, the edification of the church and so on.