Bill Clinton's approval ratings almost certainly would not be in the high 60's if he were still eligible to run in 2000. More likely they'd be somewhere around 50%, with around 42-43% disapproving.
Against W, that's probably enough for a victory in the popular vote, and by a relatively comfortable margin. The electoral college might be closer.
In this scenario, facing an incumbent President Clinton, Bush doesn't pick Cheney. I think he specifically picks a female running mate, to draw a contrast with Clinton.
I also think Gore would decline to remain on the ticket in this scenario, out of disgust for Bill's affairs, and his general treatment by Bill, and relative sidelining in favor of Hillary.
While Bill is tempted to just pick Hillary, he ultimately decides that's a bridge too far, and as IRL, Hillary runs for the Senate.
President Bill Clinton/Senator Joe Biden (D) 51%
Governor George W. Bush/Governor Christine Todd Whitman (R) 46%