A VP working in conjunction with a friendly president can have some measure of influence and public presence. A VP lacks the constitutional authority to do what they want when they want, though. Their only constitutionally mandated duty is to preside over the senate and cast a vote in the case of a tie. They have no executive powers that aren't explicitly granted on a statutory basis, and no legislative powers barring a 50-50 tie.
Considering Clinton and Warren's history, I wouldn't expect the kind of Biden Bromance we currently have in the oval office. And I certainly wouldn't expect the kind of mentor relationship Cheney had with Bush2. These are two high-profile, ambitious, successful politicians who haven't always seen eye to eye.
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