How is this baptism different?

John the Baptist separated it out before Jesus began his ministry. In John 1:29-34, John clearly says what we have experienced as reality.

"The next day he saw Jesus coming toward him, and said, "Behold, the Lamb of God, who takes away the sin of the world! This is he of whom I said, '‘After me comes a man who ranks before me, because he was before me.’ I myself did not know him, but for this purpose I came baptizing with water, that he might be revealed to Israel." And John bore witness: "I saw the Spirit descend from heaven like a dove, and it remained on him. I myself did not know him, but he who sent me to baptize with water said to me, 'He on whom you see the Spirit descend and remain, this is he who baptizes with the Holy Spirit.' And I have seen and have borne witness that this is the Son of God." ESV

Jesus explained the concept further in Acts 1:5&8 before He ascended. Not only is it different from the baptism of water for forgiveness, it has a different purpose.

"...for John baptized with water, but you will be baptized with the Holy Spirit not many days from now... you will receive power when the Holy Spirit has come upon you..." ESV

Paul explains a little in Acts 18 & 19

The end of chapter 18 talks about the powerful teachings of Apollos in which it is said, ...being fervent in spirit, he spoke and taught accurately the things of Jesus, though he only knew the baptism of John.

In 19:4-6 Paul explains it like this, calling John's baptism of water for repentance. Then he lays hands on the disciples (using the phrase baptizing in the name of Jesus) and they receive the Holy Spirit—speaking in tongues and prophesying.

It's a separate event

In Acts 8:14-17 we see Peter and John going to Samaria. These people has received Jesus and baptized in the name of Jesus, but the Holy Spirit had not fallen on any of them. Verse 17 says it this way in the RSV: Then they laid their hands on them and they received the Holy Spirit.

What is clear is simple: the baptism of the Holy Spirit is one of the things that Jesus does for us and to us.