There is a pretty clear distinction between Catholics and Protestants in that regard, though there may be Protestant denominations that do use the crucifix rather than the cross.
It seems that the main issue that Protestants today have with the crucifix is that it is specifically an image of Jesus, the Son of God. Such an image is believed to be strictly forbidden in Scripture.
You shall not make for yourself a carved image, or any likeness of
anything that is in heaven above, or that is in the earth beneath, or
that is in the water under the earth. Exodus 20:4 ESV
Throughout the Old Testament, the Israelites were distinguished from other nations by the fact that they had no idols to represent God. God gives the reason for this commandment in Isaiah, where He indicates that He is righteously jealous for His own glory and will not share it with an idol.
I am the LORD; that is my name; my glory I give to no other, nor my
praise to carved idols. Isaiah 42:8 ESV
So, Protestant churches are most often decorated with only an empty Cross that is not an object of worship, but merely a symbol of remembrance. No Protestant I know of ever prays to a cross or cares whether or not there is a cross in front of them when they pray or worship. It is wholly decorative.
In contrast, Catholic churches do often have crucifixes, which have an image of Jesus, the Son of God. Additionally, there is often an image of Mary as well. These may or may not be objects of worship, but for most Protestants, these are unacceptable due to the aforementioned commandments.
Orthodox churches are often quite replete with images of Jesus, Mary, and many of the saints. Again, I'm not sure these are used as objects of worship or not, but Protestants generally reject the use of these as they are viewed as idols--or at least could be confused as such.
As a footnote, the listing of the 10 Commandments is different between Catholic and most Protestant traditions. In the Catholic listing, idolatry is not present, but is considered as part of the first commandment, and have two commandments that mention covetousness. Protestants list the prohibition against idolatry as the second commandment and only list one commandment regarding covetousness.