Yes. A cross is simply the “t-shaped” instrument that was used for crucifixion which now, bears more of a symbol of triumph for Christians. A crucifix however, is any image of that same cross with Jesus’ body (known as the corpus) depicted on it.
Catholics are free to wear either and to have either on display in their churches although most Catholic Churches would choose to have a crucifix on display while many Protestant churches opt for the cross. In other cultures the image depicted on the crucifix may also be more vivid depicting a suffering Jesus more dramatically as well.