How has the Watergate scandal had an effect on the American government?
I think people in Congress, but also in the general public, toss the word "impeachment" around much more lightly since Watergate. Clinton was impeached by the House and acquitted by the Senate for things much more minor in terms of the law than what Nixon did. Many historians and political scientists consider the Clinton trial more of a vendetta by the opposing party that would not have been possible before Nixon's scandals.
Even today, you hear the word impeachment tossed around about President Obama, without any specific accusations of law breaking, and in the 1960s, for example, that would not have been politically possible as the popular backlash against such a move would have prevented anyone from trying it.
I also think that since Watergate there has been a continual power struggle between the legislature and the executive over executive power. 9/11 allowed President Bush to be much more aggressive and powerful than he would have been otherwise because of this struggle.
To me, there have been two major impacts of the Watergate scandal on American government.
First, I would say that the scandal (along with the Vietnam War) made the Congress much more aggressive about trying to rein in the President. It has made the Congress much more wary of presidents' efforts to get more power for themselves. This effect may be wearing off after 9/11, though.
Second, I would say that it made the party that opposes the president much more likely to use Congressional investigations (if they are in the majority) to try to uncover scandals in the administration. The opposition tries to use the investigations and any scandals they find to paralyze the administration (as with Clinton during the Whitewater/Lewinsky scandal).