Backbone is an MV* framework while jQuery is a DOM toolkit.
With Backbone, you represent data as Models, which can be created, validated, destroyed, and saved to the server. Whenever a UI action causes an attribute of a model to change, the model triggers a "change" event; all the Views that display the model's state can be notified of the change so that they are able to respond accordingly, re-rendering themselves with the new information.
While jQuery is a solid API for querying the DOM with support for event handling, deferred objects, and animations. It is best suited for things like querying the DOM, making style changes dynamically, animations, AJAX requests, etc.