The title is the question.
For example, Death Valley’s lowest point is 282 feet below sea level and Mount Whitney is 14,494 feet above sea level. Difference: 14,776 feet.
China. Plenty of spots that are at sea level and they have half of Mount Everest (opinions about Tibetan sovereignty notwithstanding).
Mount Everest is on the China / Nepal border, with an elevation of 8850 meters. The lowest point in Nepal is 70 meters, the lowest point in China -154 meters:
ETA:
As noted, China’s claims to Tibet are questionable. If you don’t accept them, Nepal probably gets the prize.
BTW, we might also note that the -154 meters for China is in the Turpan Depression, located in the Xinjiang Autonomous Region. China’s right to that bit of real estate is also questioned in some circles, though not as strongly as Tibet.
Looks like it. The second-highest mountain in the world is K2, on the China-Pakistan border (though I imagine that’s also the Tibetan part of China), at 8611 meters, and Pakistan’s lowest point is sea level.
Well, the OP seems to be answered. Shall we open it to the Top 10?
Actually, it’s in the Xinjiang Autonomous Region. -154 to 8611 still doesn’t edge out 70 to 8850.
China’s claims to Tibet are morally questionable, but if you’re asking geography questions, Tibet is de facto China, so China should get this one.
Death Valley (California) is -282 feet and Denali (Mt. McKinley - Alaska) is 20,320.
Only Denali isn’t in the contiguous United States. 
Why not argue Mauna Kea to the ocean floor (10,203 m)?
Or Mount Lamlam in Guam and the Marianas Trench (11,530 m)?
USA! USA! USA!
But, much of Alaska itself is at sea level. That still leaves a height disparity of more than 20,000 within Alaska’s borders. Surely that counts. 
McKinley’s a mere 6,194 m high.
So which country has the least difference in elevation. My entry is “Tromelin Island”, 7 meters difference.
Anyone beat that?
Well, for one, that’s hardly a “country” so much as a chunk of French territory with an airstrip and weather station.
Also, yes. The Maldives’ elevation ranges from sea level to a dizzying 2.3 meters: Maldives - Wikipedia