Plugins are functions that do something to your website. They could serve a back-end purpose and never be visible to you or the public. In the other hand, a widget usually is the representation of a plugin in the form of an object that can be placed somewhere on your website via the widgets interface under the appearance menu.

Widgets are always plugins, but plugins don’t always have widgets. A widget usually is a pretty an easy way of using a plugin. I hope this short explanation clears some things up if you’re just starting with WordPress.