Its well established that both Harry Potter and Neville Longbottom could have fulfilled the prophecy:
The one with the power to vanquish the Dark Lord approaches ... born to those who have thrice defied him, born as the seventh month dies ... and the Dark Lord will mark him as his equal, but he will have power the Dark Lord knows not ...
But why exactly did Voldemort chose Harry over Neville?
This may seem like a simple question, but it escapes my memory.
Dumbledore offers one theory at the end of Order of the Phoenix: Voldemort based his decision on blood purity, which is one of the only things he would have known about both of the boys, and which he can compare against his own:
“But he might have chosen wrong!” said Harry. “He might have marked the wrong person!”
“He chose the boy he thought most likely to be a danger to him,” said Dumbledore. “And notice this, Harry. He chose, not the pureblood (which, according to his creed, is the only kind of wizard worth being or knowing), but the half-blood, like himself. He saw himself in you before he had ever seen you.”
— Order of the Phoenix, chapter 37 (The Lost Prophecy)
J.K. Rowling confirmed in an interview that this wasn’t just a theory; this was the main reason that Voldemort went after Harry:
Why did Voldemort pick Harry and not Neville?
Dumbledore explains this in Order of the Phoenix. Voldemort identified more with the half-blood boy and therefore decided he must be the greater risk.
At the end of OotP Dumbledore says
“He chose the boy he thought most likely to be a danger to him,” said Dumbledore. “And notice this, Harry. He chose, not the pureblood (which, according to his creed, is the only kind of wizard worth being or knowing), but the half-blood, like himself. He saw himself in you before he had ever seen you.”
But this doesn't fit Voldemort's character, or more precisely it doesn't fit his situation. He has a prophecy about someone who can defeat him. Most likely it is a baby that will soon be born, and we know that he is aware of Harry and Neville, otherwise it would make no sense to speak of a choice.
It doesn't make sense for someone like Voldemort to choose one or the other instead of killing both. Even if he was 99% sure that it was one of the boys, he would still kill them both just to be safe.
If we want to call it a choice, it is based on opportunity and not preferences. The Potters were protected by a Fidelius. We can only hope that Dumbledore warned the Longbottoms and they also chose to protect themselves. Unfortunately for the Potters, the secret keeper was Pettigrew, which presented the opportunity to Voldemort to kill Harry (and his parents). Voldemort assumed that he would survive and would kill Neville at a later time, whenever he would have the opportunity.
Voldemort went to kill Harry Potter because Harry filled the requirements of the Prophecy. Voldemort also had the choice of Neville Longbottom, but he chooses Harry to kill because Harry is a Half-Blood, like Voldemort, where as Neville was a Pure-Blood. Voldemort saw Harry as a greater risk that Neville. He also chose Harry because Harry is most like Voldemort.
