I think Rent was probably most influential in it's ability to create a devoted fan base in an audience that was largely ignored by Broadway for decades (teen to early-twenties). It started the brilliant marketing device of employing the open rush policy that morphed into the now popular lottery. Remember the kids camping out on the sidewalk to purchase tickets? There was even a fashion display devoted to Rent at Bloomingdales.
As for the score being influential, just listen to The Last Five Years, Bare, I Sing, or Bright Lights, Big City. Intentional or not, it's hard to ignore the distinct style that was unique to Rent when it opened (especially in Bare, which is almost a carbon copy of much of the Rent score). I do think it opened the doors for a certain contemporary sound that had yet to find its way into the mainstream of musical theatre and has not been as successful since.
"What can you expect from a bunch of seitan worshippers?" - Reginald Tresilian
