The reason is that whether someone receives the host or the cup he receives Jesus Christ. As the Council of Trent stated:
By reason of the hypostatic union and of the indivisibility of His glorified humanity, Christ is really present and is received whole and entire, body and blood, soul and Divinity, under either species alone; nor, as regards the fruits of the sacrament, is the communicant under one kind deprived of any grace necessary for salvation.
As to why most parishes only distribute under one species is a matter of practicality. To distribute communion under both species requires extra ministers which not every parish has available. The physical layout of the aisles of some parishes is not convenient for distribution of both kinds. There is also the issue of germs during cold and flu seasons.
However, no matter how one receives communion he receives the fullness of Christ’s body, blood, soul and divinity and all graces.