Borax and washing soda are definitely not the same. Although they have certain elements in common, they have different chemical compositions. Both substances are found in nature and are processed for commercial distribution by a variety of different companies.
Although they are completely separate products, Borax and washing soda are often used in similar ways. For example, they both have a history of effectiveness as household cleaners and as laundry boosters. Each one is also well-suited to certain uses in home gardens. Borax works well as a pesticide in some cases, while washing soda helps reduce mildew on certain plants, such as roses.