I also went to the Haram al-Sharif, the sacred enclosure. They are the same place. I'd visited there before, sat in the vast expanse of the al-Aqsa mosque, said to be built over the stables of King Solomon. I visited the beautiful golden Dome of the Rock. Meaning is stacked atop meaning.
Every site has multiple names and multiple meanings, representing mutually exclusive claims.
The Temple Mount itself means the future to Jews, the present to Muslims, and the past to Christians. How far we are from that now. To Muslims it is the site of two great mosques, the religious center for Palestinian Muslims, and inevitably, a political symbol of their claims on the soil of Jerusalem, claims that cannot be argued away by rhetoric or the strong emotion on the Jewish side. Finally, to Christians the Temple Mount represents the triumph of Christianity over Judaism, the fulfillment of the prophecy of Jesus that not one stone will stand on another in that site. When I visited the site the first time, I looked in awe at the rock in the well of souls beneath the dome.