There's something about the president which clearly separates him from European prime ministers. One way to understand it is to think about the president as something like a monarch.
For most of American history the president has combined in one person both the effective and the symbolic parts of government in a way which no equivalent European high office approximates. We are used to referring to America as a "new" country, but in fact its political institutions - like the presidency - are much older than those of virtually any European country.
This antiquity, traceable back to a revered founder, contributes to its symbolic value. The president is expected to embody the nation as a whole against local or sectional interests, much like a monarch, especially as he is selected by popular vote; and this contributes to expectations of individual probity and a removal from political street-fighting which often seem bizarre or downright perverse to European eyes. The crime of a Nixon or a Clinton lies in large part in standards of behaviour which may be tolerated at lower levels but are considered below the highest office.
As the centre of symbolic power in the American political system, we find that the politics which surrounds the president are court politics: the politics of access, influence and favour. Such things are keenly tracked much more closely than in any European system. And for as long as American society was relatively cohesive, these politics sufficed and it was possible to sustain the view of one office representing the nation in itself, occasionally led astray by poor counsel; the question that recent decades seem to pose is whether, as American society seems to fragment, one man can ever again manage the competing pressures while maintaining the dignity of his office.