Well, I'm trying to get this column syndicated (and thus have a decent job), but for now, you guys get it for free.
My mother asks “Why do we have Congress instead of a parliament?”
Basically, one of the guiding principles when various people wrote the Articles of Confederation and Constitution was “don't be like England, the dirty oppressing bastards”. More seriously, James Madison and John Jay and the others did want to be different from England, both in correcting things that the Westminster system did badly and in distinguishing themselves from the monarchy. The specific idea here was the separation of powers, and the creation of distinct legislative and executive branches of government.
See, the English Parliament is tasked both with making laws and enforcing them, with the King at the time not having much of a set role —which caused about as much turmoil as you'd expect until the institution devolved into largely ceremonial duties. The framers separated those two roles: the House and Senate debate and pass laws, and the President enforces them. More accurately, the President oversees the various Cabinet departments and agencies that enforce the law, like FDA inspectors and FBI agents, but the President is the avatar of the executive branch far more than the Speaker of the House is for Congress.
Unlike Parliament, where the majority party or coalition of parties chooses a Prime Minister, who thereafter is in charge of pretty much the entire government, control can be split in our system, as it is at the moment. Democrats control the Senate and Presidency, and Republicans control the House, with cooperation between the two being required for anything to happen, including the yearly budget, without which the government shuts down. Whether or not the possibility of divided government is a good thing is up for debate (personally, I'm fine with it), but our government was deliberately set up to decentralize power, so as to prevent the rise of a new king or other unitary wielder of power.
Now, you might think “Hey, the President is a centralized power figure, right?”, and that's not unreasonable or terribly inaccurate, but there are a lot of things that the President simply can't do, like make laws or spend money (the latter power is actually only in the House). In fact, several presidents have opined both before and after leaving office that the biggest part of the job is simply persuading people.
Teddy Roosevelt even said “The Presidency is a bully pulpit”, though unfortunately teachers never explain that he frequently used 'bully' to mean 'great' or 'cool', which was slang from his childhood, and without that the quote is pretty much meaningless. He meant that the position was great for talking to people and convincing them of your position, not beating someone up or something, and really “bully pulpit” with the modern American meaning of bully doesn't make any sense.
Basically, the British Parliament is roughly equal to Congress and the Presidency, which were split up to divide power, and further structured internally to reduce the amount of power any one person could have. No one person leads Congress, and the whole institution can survive someone bad in any position, particularly since Congress elects its own leadership and is, to an extent, trusted to curb the excesses of the President, who holds more power in the executive branch than any one member of Congress does legislatively.
It is worth noting that the separation of powers is a general principle in political theory, and found in governments around the world: the basic functions of government fall into the three areas of executive, legislative, and judicial, and in most countries they are distinct from each other to some extent. The idea was first found in ancient Greece and was adopted by Rome, though the specific phrase was coined by French philosopher Baron de Montesquieu. (I'm never going to be able to spell that without looking it up.)