Earth is not a perfect sphere, and bulges out at the equator. According to the story below, the point on earth closest to the moon is the summit of Mount Chimborazo in Ecuador.
"Mount Chimborazo, in the Andes, is a 20,000-plus-foot peak sitting on top of a bulge on the Earth. Mount Everest is a 29,000-plus-foot peak sitting lower down on that same bulge. Because Chimborazo is a bump on a bigger part of the bulge, it is higher... Chimborazo is 1.5 miles higher than Everest! Or, if you will, 1.5 miles closer to outer space."
Elsewhere in the story, engineer/surveyor Joseph Senne is quoted as saying "When all the calculating is over, we think we have discovered the spot on Earth that is closet to the moon and outer space." (The next sentence: "It is Mount Chimborazo in Ecuador.") Later in the story: "But if you want to stand on the place on Earth that is closest to the moon, that would be Mount Chimborazo!"
http://www.npr.org/templates/story/story.php?storyId=9428163