Carter thought that he'd found the tomb of a king, even though all he could see was a rubble filled passageway leading to another sealed door. On November 22, he did not open that door, but rather telegraphed Carnarvon telling him to come immediately. With his patron present, Carter broke a hole in the second door on November 26.
When Carnarvon asked him if he saw anything, Carter replied: "Yes, I see wonderful things"Howard Carter knew who paid the bills, and more importantly knew why those bills got paid. Carnarvon had a sense of noblesse oblige, no doubt, the sense that the privileged class was required to give to society. Adding to the store of History, of advancing mankind's understanding of its past was a very common pastime for the nobility.
Carter knew that wasn't the real motivation. The excitement of the find was the goal, and he didn't cheat Carnarvon out of that excitement. Holding up the dig until Carnarvon could arrive from London, he gave Carnarvon the story that he would tell at dinner for the rest of his life.
Not a bad return on his exploration investment, actually.
Motivations are strange things. Such a little thing, in return for serious funding.
And because we're talking about King Tut, this is obligatory: