Catholic is the original Christianity, the Roman Christianity, the one most commonly found in Western-Continental Europe. Thanks to Martin Luther and the Protestant Reformation, there are many protestant Christians. Many different sub religions still considered Christian.
All Catholics are Christians by default, but NOT all Protestants (that you incorrectly called Christianity) are Catholics. The best thing about Protestantism is that, it denies the authority of the Catholic Pope; and the best thing about Catholicism is that, the Pope is not a Protestant.
The difference, however, is ONLY in the fact that, every new Christian sect is just a LITTLE (almost negligible) less superstitious than the old. In essence, both Catholicism and Protestantism are twins from the same mother -- vultures from the same egg!
They both consider themselves Christians. You should have said Roman Catholics and Protestants. The biggest difference between those two is that the Roman Catholics believe that the Pope and the so called "Virgin Mary " (a pagan goddess) are the mediators between man and God and the Protestants believe that Jesus Christ is the one mediator between man and God. They have both gotten so far off track that neither has much resemblance to the original.
To be Catholic is to be fully Christian. All Catholics are Christians, but not all Christians are Catholics. A Christian is someone who believes in Jesus Christ as the Son of God and is baptized. A Catholic is a specific kind of Christian - one who also believes the Catholic Church was founded by Christ and accepts it as possessing the fullness of Christian truth and the means of salvation. Being Catholic involves three main things: (1) sincerely holding to the Catholic faith as taught by the Magisterium, which is the official teaching of the body of the Church; (2) accepting the sacraments of the Catholic Church; and (3) submitting to the teaching authority of the pope and the bishops in communion with him. (My answer continues in the comments)