Cheney was practically the president during most of Bush's presidency. Biden is as influential as Nixon was during Eisenhower's administration. Remember what Ike said when asked about Nixon's experience in the executive? 'Give me a week, and I might come ip with an answer'. Ask Obama, and maybe he'll reply in a less damaging way, but the messages' essence would remain the same.
"Bush-Cheney lite," as Rose put it. The president suggests the biggest difference is that he's more thoughtful than his predecessor; he's not charging into Syria on skimpy evidence, and he's added oversight to the NSA's spy programs to protect civil liberties. "Some people say, 'Well, you know, Obama was this raving liberal before. Now he’s, you know, Dick Cheney,'" Obama quipped. "Dick Cheney sometimes says, 'Yeah, you know? He took it all lock, stock, and barrel.'"
Joe Biden is a good Vice-President and knows what he is doing, but he does not seem to have the clout that was wielded by Dick Cheney. Cheney was very involved in matters since he had experience as a Secretary of Defense previously. He was someone to whom people listened. I do not think Joe Biden has that much clout.
Dick Cheney wielded vast amounts of power for a Vice President, in fact Cheney was probably one of the most powerful VPs of all time. Biden plays a much more supporting role to Obama, and rightfully so. He has a tendency to speak out random nonsense at the oddest times, and as a result is kept restrained.
I would argue that he is not more powerful than Dick. I consider the relative impasse we've had in Congress for the past ten or fifteen years to be evidence of this. If anything, Joe has less power because the current political climate is more polarized, making bi-partisan cooperation pretty much impossible on any issue.
It could also be argued that, in the aftermath of certain rumors coming out about the Bush presidency, there is no way that Joe could really ever have that much power. The facts as I have heard them are that Dick basically ran the show until the very end of Bush Junior's second term.
Dick Cheney was one of the most "powerful" vice presidents of our time. He came from his previous position at the largest weapon manufacture company in the world. He then aided Bush in the decision to wrongfully attack Iraq, allowing his former company to post insane record profits that had never been seen before. Money controls all politics and Dick Cheney had his hands in a lot of pockets.