Harry and Cedric had equal points going into the Third Task:
"Ladies and gentlemen, the third and final task of the Triwizard Tournament is about to begin! Let me remind you how the points currently stand! Tied in first place, with eighty-five points each — Mr. Cedric Diggory and Mr. Harry Potter, both of Hogwarts School!"
Harry and Cedric touched the Triwizard Cup at the same moment:
He grabbed Harry's arm below the shoulder and helped Harry limp toward the plinth where the cup stood. When they had reached it, they both held a hand out over one of the cup's gleaming handles.
"On three, right?" said Harry. "One — two — three —"
He and Cedric both grasped a handle.
Whether the tournament is won by having the most points or by touching the cup first (as I asked in this question), it should have been a tie between Harry and Cedric — they had the same amount of points and touched the cup at the same time.
Indeed, the whole reason that they took the cup at the same time was that they wanted it to be tie:
"Both of us," Harry said.
"We'll take it at the same time. It's still a Hogwarts victory. We'll tie for it."
Cedric stared at Harry. He unfolded his arms.
"You — you sure?"
"Yeah," said Harry. "Yeah... we've helped each other out, haven't we? We both got here. Let's just take it together."
For a moment, Cedric looked as though he couldn't believe his ears; then his face split in a grin.
"You're on," he said. "Come here."
Yet Fudge gave the entire thousand galleon prize to Harry:
"Your winnings," he said shortly, taking a large bag of gold out of his pocket and dropping it onto Harry's bedside table. "One thousand Galleons. There should have been a presentation ceremony, but under the circumstances..."
This was clearly the entire prize money, as that is the amount that was specified before the tournament began:
"The heads of Beauxbatons and Durmstrang will be arriving with their short-listed contenders in October, and the selection of the three champions will take place at Halloween. An impartial judge will decide which students are most worthy to compete for the Triwizard Cup, the glory of their school, and a thousand Galleons personal prize money."
Why was Harry considered the sole winner if it was actually a tie between him and Cedric? Granted Cedric was dead, but the money could still have been given to whoever inherited the rest of his possessions.