Plain crosses and more elaborate crucifixes are both common symbols of the Catholic faith. Crucifixes bearing the image of a suffering Jesus have only been common among Catholics for about 1000 years, while plain crosses have a much longer Catholic history.
Some non-Catholic Christians wrongly consider a crucifix a violation of the Commandment against making and worshiping graven images. If you are living with such people, out of consideration of their weak consciences, a plain cross might actually be the more loving choice. See 1 Corinthians 8:1-13.