The gift of the Holy Spirit, in my opinion, means the release of the power of the Spirit. When a person is born again, they receive the Holy Spirit as the change in our human spirit. But when one is baptized in the Holy Spirit, we become permeated with the Spirit. The word baptize is Greek for dying fabric with a color. That color permeates the fabric. When you are born again you drink in the Spirit, but it is contained within you. The baptism pours through you. The end result is a magnification of faith in God. The gifts of the Spirit are simply different manifestations and functions of the power of the Spirit moving through a baptized believer.
Receiving the gift of the holy spirit is the beginning of a relationship with God when He finally comes to dwell within you and gradually reveal Himself to you through Jesus. In other words, it's only through the holy spirit can anyone have a relationship with God. "No one comes to the Father except through me." John 14:6
When God baptizes someone with his Holy Spirit they have been a gift, the Holy Spirit. Depending on God's will, he can give a person a spiritual gift, which the Holy Spirit bestows to us. To one he may give the gift of healing, to another the gift of understanding, to another, speaking in strange tongues, etc...