The empty tomb accounts
Who came to the tomb and what happened next differ — Mark ends with the women saying nothing to anyone in fear, Matthew has them run to tell the disciples; the angel count and timing vary similarly.
Mark 16:1-8
1When the Sabbath was past, Mary Magdalene, and Mary the mother of James, and Salome, bought spices, that they might come and anoint him.
2Very early on the first day of the week, they came to the tomb when the sun had risen.
3They were saying among themselves, “Who will roll away the stone from the door of the tomb for us?”
4for it was very big. Looking up, they saw that the stone was rolled back.
5Entering into the tomb, they saw a young man sitting on the right side, dressed in a white robe, and they were amazed.
6He said to them, “Don’t be amazed. You seek Jesus, the Nazarene, who has been crucified. He has risen. He is not here. Behold, the place where they laid him!
7But go, tell his disciples and Peter, ‘He goes before you into Galilee. There you will see him, as he said to you.’ ”
8They went out, and fled from the tomb, for trembling and astonishment had come on them. They said nothing to anyone; for they were afraid.
Matthew 28:1-10
1Now after the Sabbath, as it began to dawn on the first day of the week, Mary Magdalene and the other Mary came to see the tomb.
2Behold, there was a great earthquake, for an angel of the Lord descended from the sky, and came and rolled away the stone from the door, and sat on it.
3His appearance was like lightning, and his clothing white as snow.
4For fear of him, the guards shook, and became like dead men.
5The angel answered the women, “Don’t be afraid, for I know that you seek Jesus, who has been crucified.
6He is not here, for he has risen, just like he said. Come, see the place where the Lord was lying.
7Go quickly and tell his disciples, ‘He has risen from the dead, and behold, he goes before you into Galilee; there you will see him.’ Behold, I have told you.”
8They departed quickly from the tomb with fear and great joy, and ran to bring his disciples word.
9As they went to tell his disciples, behold, Jesus met them, saying, “Rejoice!” They came and took hold of his feet, and worshiped him.
10Then Jesus said to them, “Don’t be afraid. Go tell my brothers that they should go into Galilee, and there they will see me.”
What each tradition says
Rabbinic / Midrashic
The Jewish Encyclopedia states that the story of Peter recognizing Jesus as “Christ, the Son of the living God” is as mythical as the story that he foretold his crucifixion and resurrection on the third day in fulfillment of Scripture. It says that story is discredited by the disciples’ own attitude, citing Mark xvi. 8, Luke xxiv. 21, and John xx. 9.
The Jewish Encyclopedia (New York: Funk & Wagnalls, 1901-1906), s.v. 'New Testament', The Jewish Encyclopedia, s.v. 'New Testament' — section on the Gospels and the resurrection tradition (Mark xvi. 8 cited)
Patristic
Augustine offers that the women did not venture to tell either of the angels themselves, lacking courage to reply to what they heard. Alternatively, they were not bold enough to speak to the guards lying there. The joy Matthew mentions is not inconsistent with the fear Mark notes.
Augustine of Hippo (354-430), On the Harmony of the Gospels (trans. S. D. F. Salmond), in NPNF ser. 1, vol. 6 (Buffalo: Christian Literature Publishing Co., 1888) — Book III, ch. 24, §64 (on Mark xvi. 8 and Matthew xxviii. 8)
Catholic
James F. Driscoll notes that Matthew 27:56 names Mary Magdalen, Mary the mother of James and Joseph, and the mother of the sons of Zebedee. Comparing this with the parallel passage gives, he says, a well-grounded probability that Salome is identical with the mother of the sons of Zebedee.
James F. Driscoll, The Catholic Encyclopedia, s.v. 'Salome' (New York: Robert Appleton Company, 1912) — the section identifying Salome with the mother of the sons of Zebedee
Orthodox
No sourced position found. Checked the Eastern corpus in NPNF/ANF: Chrysostom's homilies on Matthew 28 (Homily LXXXIX) and John 20 (Homily LXXXVI), Gregory of Nyssa, John of Damascus, and the 1672 Synod of Jerusalem — no Orthodox text found that engages the differences between Mark 16:1-8 and Matthew 28:1-10 (who came, the angel count, the silence).
Reformed / Protestant
Calvin suggests the difference between Matthew and Mark may mean Magdalene saw Christ before the other women, or that Matthew used synecdoche, extending to all what was peculiar to one. More probably, Mark names her alone because she first saw Christ, though her companions also saw him in their order, and Matthew attributes this to all.
John Calvin (1509-1564), Commentary on a Harmony of the Evangelists, Matthew, Mark, and Luke, vol. 3 (trans. William Pringle; Calvin Translation Society) — On Matthew 28:8-10; Mark 16:8-11; Luke 24:9-12 (the women's report and the appearances)
Evangelical
Barnes suggests John mentions only Mary Magdalene because he gives a particular account of her interview with the risen Saviour. There is no contradiction among the evangelists; mentioning only some who were there does not deny others were present also, as the old maxim says, he who mentions a few does not deny more.
Albert Barnes (1798-1870), Notes on the New Testament, Explanatory and Practical (enlarged ed. by Robert Frew; CCEL carries the Baker reprint) — Note on Matthew 28:1 ('Mary Magdalene and the other Mary')