Two things about him make him both different and important in my view. One, he was really the first "progressive" President, who broke the Gilded Age cycle and took on business and social reform, and was popular enough to get away with it. Two, he was the first real conservationist in the White House, and things he did, like helping to establish national parks, and setting aside some of our natural resources were actions that would have very long lasting and positive effects on our country. Plus, you have to admit, the guy is about as interesting as a president gets.
Teddy Roosevelt exuded strength and resolve, something lacking in many leaders today. He made it clear to European powers that the United States was a power and had control of the Western Hemisphere. Unlike more modern presidents, Roosevelt was himself a veteran of battle and understood war. His institution of the National Parks was not only a means of protecting lands from exploitation by big business, but the parks preserve some of our greatest national treasures: the Grand Canyon, Old Faithful, the Badlands, the Petrified Forest, etc.
Here are some other interesting facts about Roosevelt:
Teddy Roosevelt won the Nobel Peace Prize as President when it actually meant something for mediating the Russo-Japanese War. Roosevelt garnered more third-party votes (the Bull Moose Party) than any other third-party candidate in U.S. history. He wrote 25 books himself and he was the first to dub the executive mansion "The White House."
Maybe all these and more things are why his head is carved on Mt. Rushmore.
I think some of the things accomplished by Roosevelt were important. He was instrumental in the building of the Panama Canal which provided a shorter sea route for trade. He was instrumental in conservation and was an advocate for preserving our lands. He also, as mentioned above was a big part of making certain big companies were not allowed to run amok.
I really don't think he was as important as people think he was -- I wouldn't have put him on Mt. Rushmore, for example. But here is why people would say he was important:
- In domestic affairs, he was known as a "trustbuster." He was the first of the Progressive presidents who tried to use the power of government to regulate the economy and to keep big companies from exploiting and mistreating workers. He is also known for advocating for national parks and national forests (partly to curb the power of big business to exploit the nation's resources.
- In foreign affairs, he was known for his aggressive ideas. He wanted to expand US power worldwide. This can be seen, for example, in his Roosevelt Corollary to the Monroe Doctrine and by his sending the Great White Fleet on a world tour to show US power.