Why do Catholics use a crucifix and Protestants the cross?
This is a some what complex question, but I will try to keep it simple. First, we must realize that there are many differnent branches of Protestants. What applies to the Baptist may not necessarily follow for the Anglicans. With that being said, I will talk about the Protestants in general to answer this question.
Catholics use the crucifix as a symbol because we realize that at Mass the sacrifice of Calvary is re-presented (not represented like a symbol) in a non-bloody manner. So the crucifix gives us a visible sign of what is happening on the altar. Catholics also embrace the crucifix because we believe in redemptive suffering. Because we are baptized, we are truly part of the Mystical Body and thus truly part of Christ. Therefore, when we suffer and "offer it up" we really do help sanctify ourselves and the rest of the world. This is why St. Paul says we "make up what is lacking in the sufferings of Christ"-Colossians 1:24. As Catholics, we acknowledge that suffering has value because Christ's suffering has value. It brings us eternal life. Thus, we promote the crucifix.
Protestants on the other hand, through a misunderstanding of the Mass in thinking that Catholics teach that Christ dies again and again at every Mass and that being blatent contradiction of Scripture (see Hebrews 7:27), in reaction, embraced the cross instead of the crucifix to show that Jesus only died once for our sins. The irony is that they embraced the cross instead of the crucifix to stop heresy, but in actuality the heresy of the "recrucifixion" of Christ at Mass is something that the Catholic Church does not teach and even has condemned (This by the way is part of the reason that many protestant Churches do not have a Liturgy of the Eucharist part to their Sunday services).
Protestants also generally do not see value in suffering. Some view suffering like poverty as a punishment from God. This is based on the Old Testament because when God's favor was with Israel, they were wealthy and did not have to suffer. They suffered when they were being punished. Of course, punishment is not the only reason for suffering because God was born poor and even counseled it to get to heaven. The rich young man asked Jesus, "What must I do to be saved?" Jesus answered, "Follow the commandments", then, "go, sell everything you have and you will have treasure in heaven"-Mt. 19:16-21. So we see the "health and wealth gospel" can not be a true Gospel and that suffering is not necessarily a bad thing.
Finally, because many Protestants believe in the sola fides or faith alone doctrine, they believe that redemptive suffering is a work and thus not necessary for salvation. What is important is to believe in Christ. Therefore, suffering is insignificant.
As Catholics, we believe that it is necessary to believe in Christ, but believing in Christ means to listen to his teachings as well. These teaching come from Christ and His Church. Jesus tells us that we all must "take up our cross daily and follow him"-Lk. 9:23. St. Paul tells us, "The Spirit itself bears witness with our spirit that we are children of God, and if children, then heirs, heirs of God and joint heirs with Christ, if only we suffer with him so that we may be glorified with him"-Romans 8:16-17. St. Peter even tells us that suffering is a grace that we are all called to (see 1Peter 3:19-21). So we see that suffering leads us to salvation. The cross by itself or without Christ is empty and has no meaning, but with and in Christ it gives us the deepest meaning of all, the answer to life's questions. It gives us Heaven! That is why, we as Catholics promote the crucifix and like St. Paul, "proclaim Christ crucified"-1Corinthians 1:23.