From the Mennonite Encyclopedia Volume 5. (These responses are based upon differences at the time of the Reformation in the 16th century.)

The term “Anabaptists” is the most frequently used term to describe the group that later became known as the Mennonites. The group was nicknamed “Anabaptists” because of their rejection of the validity of infant baptism and an insistence on baptizing only believers.

One of the key differences has to do with the church and its place in society. Anabaptists believe that the church and the state must be separate. Anabaptists believe that Christians have a higher allegiance or authority than the law. The state maintains the law and order in a society that is largely unchristian. Anabaptists believe that the church is to be suffering, forgiving, disciplined, loving followers of Christ. They cannot fill the law-enforcing role of the state. Mennonites take the stance of non-participation in war and believe that peace must be a way of life.

Today the Catholic Church also thinks it is important for the church to work (even against the state) on matters about war, abortion and execution, to name a few. In a situation where it looks as if war is inevitable, the Catholic Church says it is permissible to defend one’s country, friends, or oneself.

There are also some differences in the understandings about the nature of the church. Anabaptists/Mennonites believe that the church is the company of the committed, not simply those who once were baptized. The church is voluntary, adult, holy, full-time, caring, disciplined. Catholics too urge those who have been baptized as infants to confirm their own faith and allegiance to Christ once they come to an age where they are ready to make that commitment.

The Mennonite understanding of the role of Mary differs from some Catholic teachings. We believe that she was highly favored by God but we do not believe that she is an intermediary to the Father.

The Mennonite church organization is quite different from that of the Catholics; it is not a hierarchical system but rather more congregationally based. Church organization beyond the congregation includes district conferences. These are fellowships of ministers and delegates chosen by congregations.

Mennonites observe the Lord’s Supper but understand the bread to be a sign of Christ’s body and the cup as a sign of the new covenant. As Christians eat the bread and drink the cup, they experience Christ’s presence in their midst. Catholics believe in the doctrine of transubstantiation during the Lord’s Supper. Transubstantiation means believing that the bread and wine are transformed into the actual body, blood, soul, and divinity of Christ, with only the appearances of bread and wine remaining; communion is generally closed to those who are not Catholic.

Information on Catholics from http://www.catholic.com/tracts/who-can-receive-communion

More information on questions of authority of church or state here: http://www.catholic.com/magazine/articles/when-is-it-okay-to-disobey