Selenium is a browser automation tool, that's all. What you do with this power is totally up to you. Its open source and does not require any license to be used. You can customize it as per your requirement and redistribute it as well. Selenium uses JavaScript as the core. So, anything that is beyond the scope of JavaScript cannot be done using Selenium. To use Selenium for TestAutomation, you have to combine it with Testing frameworks like TestNG, JUnit in case of Java MBUnit, in the case of C#.
TestComplete is a functional automated testing platform.
TestComplete is a functional automated testing platform developed by SmartBear Software. TestComplete gives testers the ability to create automated tests for Microsoft Windows, Web, Android (operating system), and iOS applications. Tests can be recorded, scripted or manually created with keyword driven operations and used for automated playback and error logging.
-Wikipedia
TestComplete is paid software.
