Question:
Answer:
Major differences include that Messianic Jews do not believe that Jesus founded the Catholic Church as the New Covenant fulfillment of Old Covenant Israel. Consequently, they would not recognize St. Peter as the first Pope, nor the bishops as apostolic successors in general, nor the Bishops of Rome as Successors of Peter in particular.
Thus, as with other forms of Protestantism, Messianic Jews manifest the problem of authoritatively interpreting Scripture without a God-ordained Magisterium. As I note in my article, “Pentecost and the Papacy,” Catholics rely on the witness of Jesus and the early Church for definitive guidance.
Another major difference is that Messianic Jews do not recognize most of the sacraments, including the priesthood and the Eucharist. They would see their communion services as a mere commemoration of what Jesus did at the Last Supper. Catholics realize that the Sacrifice of the Mass make present the one Sacrifice of Calvary, which culminated in everlasting glory in the heavenly sanctuary, and definitively fulfilled the Old Sacrifices of Passover and Yom Kippur.
For more on how the early Church believed in the Eucharist, see this article.