This is quite clearly covered in both the novel and the film.
The wand is very very (very) dangerous and using it has led to the messy downfall of every wizard who's ever possessed it along with large numbers of civilian casualties. Since he just made a great big speech in front of the entire wizarding world that the wand he just won is the Elder Wand, frivolously wandering around repairing things and daring every would-be Voldemort wannabe to challenge him for it would be a bad idea.
Oh, and he's a bit tired too
After a while, exhausted and drained, Harry found himself sitting on a
bench beside Luna.
In the book he decides to stuff it underground:
‘That wand’s more trouble than it’s worth,’ said Harry. ‘And quite
honestly,’ he turned away from the painted portraits, thinking now
only of the four-poster bed lying waiting for him in Gryffindor Tower,
and wondering whether Kreacher might bring him a sandwich there, ‘I’ve
had enough trouble for a lifetime.’
and in the film he decides to snap it to prevent anyone else getting their grubby mitts on it:
Hermione raises her hand and Ron stops, sees that her eyes are fixed on
Harry, who is staring at the wand, his concentration total. She
studies his face for a long moment, and then -- as if coming to
understand something-- begins to nod. As Harry peers down at his
hands, Ron and Hermione do as well, watching as he takes the ends of
the wand and...Snap