They aren't particularly big, as states go. If they were combined into one state they would still be smaller than a lot of states, and their population has always been pretty low. Why were two separate states carved out of that territory when there could have easily been one state called Dakota?
For one, your assessment of their size is a bit off. They are larger than 30+ other states, and if combined would be the 4th largest, ahead of Montana. But to the question, the original territorial capital was Yankton, in the southeast corner of now-South Dakota. When gold was discovered in the Black Hills in 1874 (See Custer and Little Bighorn), thousands of settlers poured in. Into the 1880s, railroads also started to emerge, and Yankton was considered too far from the location of many settlers post 1883, and so the capital was moved to Bismarck. By 1889, when the territory applied for statehood, the centers of population were in the northeast area of what is now North Dakota, and the southeast area of now-South Dakota. Also playing a role was Congress. The Dakota Territory heavily favored the Republican Party, and dividing them into two states would mean two extra Republican Senators, plus it would benefit the settlers in each state who would be closer to their respective capital, so it was divided into two states.