For a long time, the South Pole was one of the last great targets of explorers on Earth. There were a number of expeditions to explore the region, but for some reason, everyone who tried to reach the pole did so from the Ross Sea (towards New Zealand), rather than from the Weddell Sea (towards the South Atlantic).
Why did all the South Pole attempts use the Ross Sea instead of the Weddell?
Comintern's answer is substantially correct, but hindsight helps us identify a lot of the issues that made the Ross Sea a good idea. From a contemporary perspective, there's one really substantial factor: the Ross Sea area was relatively well understood at the time of the major 'Heroic Age' expeditions in the early twentieth century, while the Weddell was more or less a blank.
The Transantarctic Mountains overlooking the Ross Ice Shelf had been reached, and a series of expeditions culminating in the 1907–09 Nimrod expedition had shown that there was a passage through them to the polar plateau. By comparison, the Weddell Sea area was very poorly understood. The northern part of the Antarctic Peninsula ('Graham Land') had been explored, but its eastern coast was a complete mystery, and only one expedition (in 1904) had ever managed to reach Coats Land on the eastern side of the sea. The Filchner Ice Shelf was not discovered until 1912, and the Ronne was first sighted in the 1940s (!).
This map (showing the situation as known in 1911; source) demonstrates the problem quite clearly:
To an expedition planner at the time Amundsen and Scott were setting out, there was a reasonably clear route through from the Ross Sea, with fairly well-understood sea ice conditions, sites for a base camp, etc. On the other side, there was a sighting of a rocky coastline, an unexplored hinterland, and very heavy ice to get there. With the information available at that time, the Weddell approach wouldn't seem to offer any benefits over any other sector of coastline.
3 reasons that I know of - and the first is visible on your map:
Image credit NASA
Take a look at the topographical lines around Coats Land - the ascent to the Antarctic Plateau is more difficult from the Filchner-Ronne Ice Shelf than it is from the Ross Ice Shelf. It doesn't look that bad on the map, but there is roughly a 10,000ft (~3000m) elevation difference from the ice shelves to the Plateau.
The second reason was that the Weddell Sea is extremely inhospitable compared to the relative calmness of the Ross Sea. To borrow Thomas R. Henry's quotation from Wikipedia:
The Weddell Sea is, according to the testimony of all who have sailed through its berg-filled waters, the most treacherous and dismal region on earth. The Ross Sea is relatively peaceful, predictable, and safe.1
In fact, Ernest Shackleton's Endurance Expedition in 1914 failed largely due to the conditions in the Weddell Sea, and they never even made landfall on the continent.
Finally, there are 2 passes through the Transantarctic Mountains that make the approach much easier from the Ross Ice Shelf, as illustrated by Gordon Home's map of the routes followed by Roald Amundsen and Robert Scott. Amundsen's route is on the left, Scott's on the right.
Image cropped for clarity, original available here.
Note that Amundsen's ascent to the Antarctic Plateau is also about a quarter of the distance to the Pole than route from the Filchner-Ronne Ice Shelf. This also made logistical support (laying supply depots) much easier, as well as limiting the distance that had to be travelled at the higher, colder, and windier elevation.
1Henry, Thomas R. (1950), The White Continent: The Story of Antarctica
