Of the choice between Leon and Claire: you get largely the same story but many differences. Early portions mirror each other such as the gas station and the fire escape. Some are branching points like penetrating the watch room or the chief's office. Things also take different paths towards the end with Leon getting embedded in the Umbrella mystery, and Claire in the quest to shave Sherry.
But somehow, I think the scene at the end of the sewer segment captures it best:
Leon's time is mostly spent being a rookie man of mystery focused on Racoon City. And perhaps nearly getting eaten because he's more interested in flirting with the pretty girls than fleeing zombies. At best it can lead to Umbrella being exposed but it's all rather fleeting. But it's more exploratory than passionate.
Claire's a mission about Sherry. While there's much less intrigue without Ada, I feel the focus on the Birkin family makes a far more personal connection. We all know that Racoon City is doomed, 'cuz that's just how these things go. Plus let's be honest: the original game ended with a get to da choppa / nuclear kabluuy kind of end to the Spencer mansion. There's no way you're saving Racoon City, and no point in trying. But Sherry—that's a difference you can make in how the disaster plays out.
There's also how the two characters are introduced. Leon is driving into the city because things have gone dark, and he's supposed to be starting work at the police station. Claire shows up headed there in search of her brother, one of the original game's S.T.A.R.s team protagonists. You could say these kind of metrics largely keep rolling through how their stories diverge.
So in the end, I reckon the question of which character's story should you take, is the question: are you curious or do you care? Personally I feel that Claire's route makes for a better game, and Leon's for a better novelization.