Without the Holy Spirit, we are powerless. We cannot do the work of the Gospel unless we are empowered. That empowerment comes when the Holy Spirit "come upon" us. By going their separate ways, the disciples will face continued discouragement from those who crucified Christ. They need to be together to encourage each other. Also, as we see later there are important preparations to be done for the work before them, like electing a 12th disciple.
The coming of the Spirit is a filling because the believer suddenly realizes that a void has been filled. Something missing is suddenly made available. Those lacking courage, become courageous. The fearful suddenly becomes bold. The stammerer becomes eloquent. The coming of the Spirit rectifies a deficiency necessary for evangelism.