Yes, 中国 (traditional Chinese) and 中國 (simplified Chinese) are two different written representations of the same thing.
People in Mainland China usually write in the simplified form, and people from Taiwan write in the traditional form. People who write in different forms of Chinese can still communicate orally, but it can take some effort to get used to reading the other version. There more foreigners learning simplified Chinese nowadays because it is most used Chinese version in the world. Regardless what version you choose, I hope you will enjoy learning chinese, 加油!!