The United States is a constitutional republic with all of the executive power vested in the executive branch, which is headed by the President. The President and Vice President are chosen by the citizen voters—although technically electors cast the actual votes.
In a parliamentary system, the Prime Minister is the leader of the party that wins the most seats in the House of Commons and is appointed by the monarch. So the voters in the UK only vote for the Members of Parliament (MP), not the Prime Minister. Therefore, the voters in the US have more say-so over their head of government. This underscores the fact that in the UK the monarch is the sovereign while in the US the People are the sovereign.
While the election and duties of the President are spelled out in the Constitution, the duties of the Prime Minister are conventions that have evolved over time and are not spelled out in black and white. Presumably those duties could be altered, absent the will of the electorate. The US President is also the Commander-in-Chief. In the UK the Commander-in-Chief is the monarch.
While the President has authority to execute the laws, their party platform can be helped or hindered by Congress, which makes the laws. The President does have limited executive authority that can modify existing laws or create new regulations that often act like laws.
The Prime Minister, on the other hand, plays a stronger guiding role in lawmaking, and powers are not separated in this parliamentary system. While in the United States different factions at different times can stymie a legislative agenda, what results is that laws that do pass in the US usually have some measure of bipartisan consensus.
It seems that in the parliamentary system one side a has significant advantage over the other, but this advantage can change very quickly if the Prime Minister loses support or if the opposition calls for a vote of no confidence or a general election. The notion of an entire government changing very quickly is foreign to the US voter, because even if a President is removed from office, the Vice President will take their place and maintain some continuity.
While this one-sided advantage may be preferable for an idealogue, in general the slow evolution of laws and societal changes provides more stability.