I am going to assume that you are asking about George W. Bush's presidency, not that of his father.
Since you ask about the "political" features of these presidencies, I would argue that the major difference between the two is that Clinton had to deal for much of his time in office with a Congress controlled by the opposing party while Bush had a Congress controlled by his own party for most of his presidency. This was important politically because it forced Clinton to be moderate while allowing Bush to take a hard line. Clinton worked with Congressional Republicans and passed laws (such as welfare reform) that had broad bipartisan support. Bush, by contrast, had no need to compromise with Democrats and therefore pushed a much more ideologically pure agenda.