Hey Bear,
I see jQuery UI playing a complimentary role with other UI frameworks. Although there is certainly overlap between what Bootstrap and jQuery UI do, there are a lot of widgets jQuery UI provides that Boostrap doesn't—for instance Bootstrap doesn't have a datepicker.
But more importantly, there's no reason you can't use Boostrap and jQuery UI side by side. There's even a project that integrates the CSS of the two projects so you can use jQuery UI widgets with Bootstrap's CSS:
http://jquery-ui-bootstrap.github.io/jquery-ui-bootstrap/components.html.
In a more general sense, jQuery UI tries to provide extremely high quality, extensible, well documented, and accessible widgets. They're widgets that you can drop in and feel confident that they'll just work—even for users on assistive technologies (i.e. screen readers).
That being said jQuery UI only provides 12 widgets, and most developers are going to need more to build today's client-side-heavy applications. So in jQuery UI we try to make it easy to drop a widget into an existing system. If you have a Bootstrap site you can drop in a jQuery UI datepicker. If you're using Foundation you can drop in a dialog and an autocomplete. The default jQuery UI theme is purposely a grayish color so it doesn't clash with existing themes. The same methodology applies to MVC frameworks. Using Ember? Backbone?
Angular? At some point you're still probably going to need an autocomplete or dialog. We try to provide high-quality solutions that integrate into these scenarios.
The AMD support we added in jQuery UI 1.11 helps to complete this story, as it's now super simple to pick out the one or two widgets you need and get a lightweight file you can drop into your existing site.
It's also worth noting that you can use the jQuery UI widget factory to build UI components specific to your application, your company, or whatever. There are lots of people that take the 2k widget factory file from jQuery UI and forego the rest of the library altogether. If you've decided to use jQuery, we feel that the widget factory is the single best way to build reusable components on the web.
TJ