In the final battle between Harry and Voldemort, Harry apparently wins because the wand Voldemort is using is actually under Harry's allegiance:
Harry saw Voldemort's green jet meet his own spell, saw the Elder Wand fly high, dark against the sunrise, spinning across the enchanted ceiling like the head of Nagini, spinning through the air toward the master it would not kill, who had come to take full possession of it at last.
And Harry warned Voldemort of this just moments earlier:
"But you're too late," said Harry. "You've missed your chance. I got there first. I overpowered Draco weeks ago. I took this wand from him."
Harry twitched the hawthorn wand, and he felt the eyes of everyone in the Hall upon it.
"So it all comes down to this, doesn't it?" whispered Harry. "Does the wand in your hand know its last master was Disarmed? Because if it does... I am the true master of the Elder Wand."
And we see that Voldemort did for a moment entertain the possibility that he was not actually the master of the wand:
"The true master of the Elder Wand was Draco Malfoy."
Blank shock showed in Voldemort's face for a moment, but then it was gone.
- Harry Potter and the Deathly Hallows
Now there was something very simple Voldemort could have done, and if there was even a slight chance that he would otherwise lose we would expect him to do this. He could have simply pocketed the Elder Wand and switched back to his old wand. That wand had never been won by anyone else so he could be sure of its true allegiance. And he wouldn't have to worry about anything to do with the connection between that wand and Harry's wand, because he knew that Harry wasn't using the phoenix wand.
So why didn't Voldemort just switch back to his old wand once he found out that Harry might be the master of the Elder Wand?
And if, for whatever reason, he happened not to have his old wand with him, why wasn't Harry at least concerned that he might have it?