What are the claims of authenticity of these two sites of Jesus' burial, the Garden Tomb and the Church of the Holy Sepulchre?

The Church of the Holy Sepulchre is located at the spot where in the fourth century Helena, Emperor Constantine's mother, identified the very place where Jesus had been buried on Mount Golgotha. The location was upon the site of a Jewish burial chamber and beneath a Temple of Aphrodite.

Michael Grant, in The Emperor Constantine, pages 202, says that Eusebius tells us, in Life, that Helena was shown the correct location in a vision. Constantine himself ordered Bishop Macarius of Jerusalem to build on the site “a basilica more beautiful than any on earth”, without sparing money, craftsmen, labourers or materials.

Wikipedia explains that Constantine's church was built as two connected churches over the two different holy sites, including a great basilica, an enclosed colonnaded atrium (the Triportico) with the traditional site of Golgotha in one corner, and a rotunda, called the Anastasis, which contained the remains of a rock-cut room that Helena and Bishop Macarius identified as the burial site of Jesus.

Possible reasons to doubt the authenticity of Helena's claim include that building the basilica required the destruction of an important pagan temple, and the unlikelihood that Joseph of Arimathea, a wealthy and influential Jew, would choose to have the tomb intended for his own burial so close to where the Romans executed common criminals (although John 19:41 places them in proximity). Helena had been phenomenally successful in Palestine, and Grant reports the words of a scholar that her thrilling discoveries were made “with miraculous aid seldom now vouchsafed to archaeologists”.

During the post-Reformation era there was an increase in doubts regarding the traditional holy places and various other locations began to be put forward. In Aramaic, Golgotha actually means 'skull', and one of the proposed locations was 'skull hill', a small hill that resembles a skull in appearance. This contains a few natural cavities as well as a man-made cave, which Christians call Jeremiah’s Grotto and which biblical scholar Otto Thenius suggested was the tomb of Christ.

General Gordon accepted that skull hill must be Golgotha, but proposed that Jesus was actually buried in one of a number of tombs found nearby. This particular tomb also has a stone groove running along the ground outside it, which Gordon argued to be a slot that once housed a stone, corresponding to the biblical account of a stone being rolled over the tomb entrance to close it. An ancient wine press and cistern have been cited as evidence that the area had once been a garden, and the somewhat isolated tomb adjacent to the cistern has become identified as the Garden Tomb of Jesus.

John McRay (Archaeology and the New Testament, page 212,) says that two tomb inscriptions found nearby in 1885 and 1889 were used improperly, if not fraudulently, as supporting evidence for this as the site of Jesus' tomb. On its own, the 1889 inscription simply marked the site of "Deacon Nonnus Onesimus of the the Holy Resurrection of Christ and of this monastery." Jerome Murphy-O'Connor (The Holy Land, page 161) says that on this evidence, the Anglican Church committed to the identification of the Garden Tomb but subsequently removed its formal support.

The site of the Garden Tomb has really been built on speculation: that 'skull hill' really was Golgotha, that the Garden Tomb's Golgotha is north of Mount Zion and that the wine press and cistern were sure signs of the area once having been a garden.

As with the Church of the Holy Sepulchre, there are good reasons to doubt the authenticity of the Garden Tomb. Christian and pre-Christian Temple Judaism traditionally maintained the name of Golgotha as the final burial site of Adam's skull and bones, and 'skull hill' played no historic part in Jewish or Christian tradition regarding Golgotha's location. The groove could not have held a rolling stone and it has since been identified as a watering trough for horses. The cistern, which was built as part of the same stable complex as the groove, must date to the time of the Crusaders. The hill now known as Mount Zion has only had that name since the Middle Ages, before which Mount Zion referred to the Temple Mount, due east of the traditional site, but south south east of the Garden Tomb.