The Holy Spirit is the same God regardless of time. Therefore, His divine nature doesn’t change no matter whether we read about Him in the Old or the New Testaments. However, it is true that He worked differently in the Old and the New Testaments by God’s providence in order to save mankind from their sins.
In the Old Testament, God poured the Holy Spirit on to men of God by special methods in order to speak His words, to show His will through wonders, and to do His work. For example, the Spirit of the Lord began to move upon Samson the Judge, thereby doing many mighty works through him (Judges 13:25, 14:19). In other words, the Holy Spirit came on elected people restrictively at the time of the Old Testament.
However, at the time of the New Testament, by designating the Day of Pentecost as the starting point for the coming of the Holy Spirit, God sent the Holy Spirit to every saint who has received the forgiveness of sins through their faith in the gospel of the water and the Spirit. And He permits the Holy Spirit to dwell in them forever.
Therefore, after the apostles were filled with the Holy Spirit on the first Pentecost after Jesus Christ’s resurrection, all the righteous whose sins were forgiven by believing the gospel of truth can have the indwelling of the Holy Spirit (Acts 2:38). Peter went into the house of Cornelius, a Gentile and centurion of Rome, and preached the gospel of Jesus’ baptism and His blood on the Cross. While Peter was speaking the gospel, the Holy Spirit fell upon everyone who heard the word (Acts 10:34-45). This proves that the moment one hears and believes in the gospel of Jesus’ baptism and His Cross, which Jesus has fulfilled, he receives the Holy Spirit as a gift.
God caused the Holy Spirit to dwell in all the righteous who were forgiven for all their sins by believing in the true gospel. The Holy Spirit in the Old Testament played the role of leading people to Jesus Christ, and the Holy Spirit in the New Testament bears witness to God’s righteousness and stands as the guarantee for it. God’s righteousness means that Jesus forgave all the sins of the world through His baptism and His blood on the Cross. And the Holy Spirit stands as a guarantee of the gospel of salvation and helps everyone believe in it.