The major difference between these two types of government is that the American system has separation of powers while the British system does not.
In the British parliamentary system, the legislative branch is the only independent branch of government. The executive is not elected separately as it is in the United States. Instead, all of the highest members of the executive are members of Parliament as well. This means that Parliament controls the executive completely.
In the American presidential system, there are separate branches of government with distinct powers. The President and his (or someday her) top officials are not members of the legislative branch. They do not have to answer to that branch. Similarly, the legislature is not subordinate to the executive.
In practical terms, a major difference between the two systems is that the American system allows for divided government while the British system does not. In the parliamentary system, there cannot be one party in charge of the executive while another runs the legislature. The current situation in the US, with a Democratic president and a House of Representatives run by the Republicans, is not possible in the British system.