Clinton spoke at a National Urban League leadership conference north of Miami, where Bush took the same stage nearly an hour later but declined to respond in kind to the Democratic frontrunner's remarks.
The two most high-profile 2016 candidates addressed the delicate issues of structural poverty, race in America, and income and education inequality.
Clinton used a Bush slogan -- "Right to Rise," which is also the name of the leading political action committee backing Bush's candidacy -- to castigate him for shortcomings of his tenure as Florida governor.
"I don't think you can credibly say that everyone has a right to rise and then say you're for phasing out Medicare or repealing Obamacare," she added.
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"They can't rise if the minimum wage is too low to live on. They can't rise if their governor makes it harder for them to get a college education. And you can not seriously talk about the right to rise and support laws that deny the right to vote."
Clinton went considerably further than Bush in highlighting the role race still plays in determining "who gets ahead in America and who gets left behind."
While that is partly a legacy that stretches back centuries, "it is also because of discrimination that is still ongoing," she said, as she recalled the names of several young unarmed black men recently killed in police-involved shootings across America.
"These names are emblazoned on our hearts," she said.
Bush spoke far less about the crisis involving the policing of American communities and more about his revolutionising Florida's school system including launching charter schools.
And he gave a nod to President Barack Obama's efforts to bring healing to a nation grieving over deadly racist attacks.
"When Obama says that, for too long we've been blind to the way past injustices continue to shape the present, he's speaking truth," Bush said.