A "component" is a part of a system. Bower is package manager.
Which one is better? Read the Bower FAQ for differences.
Bower seems to make better what it aims for, that is a package manager for the web.
There are also Jam, Volo, or Ender. And of course, NPM.
For popularity and adoption check, you can check Ohloh
UPDATE:
Component package manager https://github.com/component/component is also a package manager for the web. It uses GitHub as a back-end database for modules, so modules that are not on GitHub will not be listed. This approach however makes things easier, as there is no need for additional credentials; authors just use GitHub.
Personally, I think Component should adapt a distinctive name, so that it would be not confused with the general word "component."