Amundsen's victory was front page news around the world. See the New York Times treatments from March 8th and 9th, 1912 below. Amundsen actually reached the pole 3 months earlier, in mid-December, 1911, but then he had to sail halfway across the world to let anyone know about it. News of Scott's death wouldn't come for several more months.
Scholars (and everyone else) point to January 3rd, 1912 as the date of Scott's fatal mistake. A decision that would lead five men to their deaths. (Scott made a lot of fatal mistakes but this one was probably the worst.)
After a month on the barrier, and weeks climbing up the Beardmore Glacier, Scott stands on the polar plateau, at the start of the final leg of the journey. Scott's diary for January 3rd reads - "Within 150 miles of our goal. Tonight I decided to reorganise, [...] Bowers is to come into our tent and we proceed as a 5-man unit to-morrow."
At that time, polar journeys were only made possible by strategically placing food and supplies in depots along the route ahead of time. Scott's journey consisted of a complicated series of depot journeys and support parties all coordinated to place a 4-man team on the plateau with enough food and supplies to complete the final dash to the pole. So after 2 years of calculating, portioning, and depoting food and supplies for 4 men, Scott decided, literally at the last minute, to take a 5th man to the pole: Henry Robertson "Birdie" Bowers.
Huntford speculates that Bowers was added to the polar party to serve as navigator, as none of the 4 original members were capable of astronomical navigation; an unbelievable oversight on Scott's part considering walking on the featureless plateau is akin to sailing on the ocean. For comparison, Amundsen brought 4 navigators to the pole.
To his credit as navigator, Bowers found the South Pole.
Scott and Amundsen took different, but very similar, routes to the South Pole. Their bases were both on the edge of the Ross Ice Shelf, which they called the "Great Ice Barrier." Most expeditions used to start here because the Barrier is a huge floating ice shelf. It's perfectly flat and easy to travel on. It's basically a giant, frozen bay that cuts deep into the continent, giving explorers a shortcut to the Pole. It's on the bottom of the map below. White is land, gray is sea ice.
Approaching the edge of the Barrier was like sailing into a 100 foot wall of ice.
Amundsen and Scott landed on opposite sides of the Barrier and, more or less, walked due south toward the Pole. Amundsen's route was about 100 miles shorter, but it was uncharted. Scott's route was longer, but had already been mapped, removing much of the danger, and equally as important, the anxiety of the unknown.
Below, Amundsen's route is on the left and Scott's is on the right. The Barrier, in gray, would be open ocean if it wasn't frozen solid. Where the Barrier meets the land stand the towering Transantarctic Mountains with the smooth high plain of the Antarctic Plateau behind.
The journey itself consists of three parts - the long, relatively easy trudge on the flat Barrier (~400 miles), the crossing of the Transantarctic Mountains (~100 miles), and the high-altitude Plateau journey (~300 miles). The crossing of the mountains is the shortest, but most dangerous leg.
Rather than climb the mountains, Scott and Amundsen used giant glaciers as their "roads to the pole."
Scott's Route: The Beardmore Glacier, previously mapped and climbed by Ernest Shackleton.
Amundsen's Route: The Axel-Heiberg Glacier. Never before seen by human beings.