Members of the International Community of Christ wear a Cross.  Members of some other Churches wear Crucifixes.   
The Crucifix represents the death of Jesus.
A Cross is a symbol of the Logos and divine Light.
The Egyptian Ankh is a variation of the cross and has the same meaning.

Until approximately 500 years ago, the Popes wore a cross, not a crucifix.

The cross was used as a symbol of God, Logos or God's Grace in many civilizations including the Essenes of Israel, the Theraputae of Egypt and the Chachapoyas or of Peru.