A subsidiary is a company that is controlled by its parent company. Even though a subsidiary works on its own, and its purposes are different from that of its parent company, the parent company is in control of its subsidiary.
A company's subsidiary is different from a company's division because a division works together with its parent company. A division's goal is usually similar to that of its parent company, and the division has to use its parent company's name. A subsidiary can use a name different from that of its parent company.
For example, YouTube is a subsidiary of Google, because Google is a search engine, while YouTube is a video sharing website. Google Videos is a division of Google, because Google Videos is a search engine that searches videos on the Internet. This is to match its parent company Google as a search engine. Also, YouTube has an entirely different name and logo from Google's, while Google Videos, as a division, must use Google's name and logo.