The Holy Spirit has always been active in the world. He has always sustained creation and natural life, renewed hearts, granted spiritual understanding, and bestowed gifts for service both in leadership and in other ways. The shift from the Old to the New Covenant did not change the Holy Spirit's role in these activities. However, the shift to the New Covenant did expand the breadth and depth of His ministry.
A major difference between the Old and New Covenants concerns the location of God's glorious presence. While the Scripture affirms the fact that God is everywhere present (i.e. Jeremiah 23:24; Psalm 139:7-10), the Scripture also affirms God's essential presence among His people. In Exodus 34:35, we are told that "the glory of the Lord filled the tabernacle" after it was constructed and consecrated. When God's presence filled the tabernacle, He did not cease to be everywhere else. Imagine sunlight lazering through a magnifying glass. The concentration of light is intense and peculiar; but, the sun does not cease shining everywhere else.
Ezekiel prophesied a day when God's glorious presence would depart from the Temple as a result of Israel's persistent infidelity (Ezekiel 10). God's glorious presence would not return until the advent of Jesus Christ. In the New Testament, we find God's essential presence in Jesus of Nazareth. In John 1:14, we are told that "the Word became flesh and dwelt among us, and we have seen his glory, glory as the only Son from the Father, full of grace and truth." The langue of verse 14 echoes loudly the language of Exodus 34:35, particularly the term "dwelt" which could be more precisely translated "tabernacled." God tabernacled in the person of Jesus. Later, Jesus refers to Himself as the true Temple (John 3:19-22).
Moreover, as God's glorious presence shifted from the Tabernacle/Temple under the Old Covenant to the person of Jesus at the start of the New, so too did the concentration of worship. Jesus spoke of a day when the worship of God's people would no longer center on the physical locale of the Temple (John 4:20-26). Once the New Covenant was ratified by the death and resurrection of Christ a new era would arrive with the outpouring of the Holy Spirit at Pentecost (Acts 2). Now, God's glorious presence resides within all of God's people indiscriminately, simultaneously, and permanently (Acts 2:16-21).
Those who turn from sin and trust in the gospel are filled with the Spirit and now become themselves God's temple: "Do you not know that you are God's temple and that God's Spirit dwells in you?" (1 Corinthians 3:4). This is true for every believer regardless of race, gender, or role within the Church.
Moreover, under the Old Covenant, God often foreshadowed privileges that would be fully realized under the New Covenant. We see this in the sacrificial system that prepared the way for us to process the death of Jesus Christ. We also see this in the few times God's Spirit indwelled a leader or a prophet for the sake of God's redemptive purposes. Two examples of individuals who seemed to have been indwelled by the Holy Spirit are King Saul (1 Samuel 10:6-13) and King David (1 Samuel 16:13). However, their experiences were unique and many scholars consider their indwelling to have been marked by a conditional permanence. The Holy Spirit departed Saul as a result of his infidelity in a similar way as He departed the Temple as a result of Israel's infidelity (1 Samuel 15). The same may have happened to David had he not repented of his murderous adultery and prayed, "Cast me not away from your presence and take not your Holy Spirit from me" (Psalm 51:11).
Under the Old Covenant, the Holy Spirit primarily facilitated a ministry of anticipation. The Holy Spirit delivered the gift of faith in God's promises to the hearts of God's people. He whet their appetites for the coming of the Messiah and the blessings He would bestow, turning their gaze mainly towards the future. At times, the Holy Spirit's manifestation would be so distinct that He aroused a heightened sense of longing for His presence and power (Numbers 11:29).
Under the New Covenant, the Holy Spirit facilitates primarily a ministry of application. He continues to deliver the gift of faith in God's promises to the hearts of God's people. However, he turns our gaze primarily to the past as we trust in all that Christ Jesus has done recognizing that "all the promises of God find their Yes in him" (2 Corinthians 1:20). The Holy Sprit applies God's promises to our lives in transformative ways. The indiscriminate, simultaneous, and permanent outpouring of the Holy Spirit upon all God's people is a privilege flowing from the Messiah's atoning work.
Much more could be said about this topic. If you would like to research further, I recommend reading J. I. Packer's Keeping in Step with the Spirit and Sinclair Ferguson's The Holy Spirit.
Thanks for asking!