Both religions are rooted in vast webs of oral tradition that were eventually (and selectively) transcribed into "sacred texts." The notion that Christianity is historically accurate is a fallacy promulgated by Christian apologists who cherry-pick "history" and propose nonsensical "this or that" arguments designed to trick people into believing in what they're selling.
I have a brilliant friend who, sadly, was recently hoodwinked by "historical accuracy" into becoming a Christian. It was as though he checked his brain at the door! Key to his buying into the Christian faith was the ridiculous argument that "Jesus was either who he said he was, or else he was a madman." He also said that there is too much historical evidence of the miracles Jesus performed to deny any of it.
I choose to allow for the fact that my friend needed something he's now finding through Christianity and I will not debate with him. If he tries to convince me of his beliefs I will politely excuse myself from the conversation or change the subject. I could ridicule his basis for his new beliefs but that would do neither of us any good. If it makes him happy to have "a relationship with Christ" then I'm happy for him.
One of the major differences between Christianity and Hinduism is that the latter doesn't propose that all human beings must believe as Hindus believe in order to avoid the eternal punishment of hell. Christianity and Islam have built-in beliefs that lead people to proselytize, so they have full-time apologists who manipulate historical evidence and religious texts for the purpose of convincing other that they are right. Any semblance of "historical accuracy" is manufactured by these concept-marketing evangelists.
This thread's title question doesn't make sense. Accurate relative to what? Religions aren't even internally consistent. Christianity's claims to historical accuracy are based on forcing history to conform to belief. There is no basis for believing if one sincerely investigates what little we know about the history of Jesus or his followers. Most of that history was censored by the Romans when they decided what would be included in the Bible. Silencing one's detractors through censorship and coercion doesn't lead to historical accuracy, it leads to mass deception.