The POTUS is the "face" of the United States throughout the rest of the world. More importantly, his appointees play significant roles in the operation of the government and the direction it takes. Functions such as Treasury, State and Defense are particularly important. Those departments are run by presidential appointees.

Domestically, the POTUS can serve in an important leadership role (something I've been very disappointed in from President Obama). You can't expect, and will never get, "leadership" from the Congress, they are a law-making body. While both the POTUS and members of the House and Senate are elected from political parties, members of Congress stay very close to their political roots in what the say and do. The POTUS on the other hand should be expected to speak out to the country and for the country, both to the electorate as well as the Congress itself. He has an agenda and priority setting role that is very important. And the way the POTUS explains and sells his agenda to the people can be very important in coalescing the country and directing the Congress towards proper governance.

There's no need getting into examples of presidents who were effective at providing such leadership and direction other than naming a few names, it's so obvious. Compare Reagan, Kennedy, Clinton to a few others: Carter, Bush 41, and Obama. The effect that the POTUS can have on the country is pretty apparent.