Both do the same work as they are used for routing purposes in SPA(Single Page Application).

1. Angular Routing - per $routeProvider docs

URLs to controllers and views (HTML partials). It watches
$location.url() and tries to map the path to an existing route
definition.

HTML

<div ng-view></div>

Above tag will render the template from the $routeProvider.when() condition which you had mentioned in .config (configuration phase) of angular

Limitations:-

• The page can only contain single ng-view on page


• If your SPA has multiple small components on the page that you wanted to render based on some conditions, $routeProvider fails. (to achieve that, we need to use directives like ng-include, ng-switch, ng-if, ng-show, which looks bad to have them in SPA)


• You can not relate between two routes like parent and child relationship.


• You cannot show and hide a part of the view based on url pattern.

2. ui-router - per $stateProvider docs

AngularUI Router is a routing framework for AngularJS, which allows
you to organize the parts of your interface into a state machine.
UI-Router is organized around states, which may optionally have
routes, as well as other behavior, attached.

Multiple & Named Views

Another great feature is the ability to have multiple ui-views in a template.

While multiple parallel views are a powerful feature, you'll often be able to manage your interfaces more effectively by nesting your views, and pairing those views with nested states.

HTML

<div ui-view>

    <div ui-view='header'></div>

    <div ui-view='content'></div>

    <div ui-view='footer'></div>

</div>

The majority of ui-router's power is it can manage nested state & views.

Pros

• You can have multiple ui-view on single page


• Various views can be nested in each other and maintained by defining state in routing phase.


• We can have child & parent relationship here, simply like inheritance in state, also you could define sibling states.


• You could change the ui-view=some of state just by using absolute routing using @ with state name.


• Another way you could do relative routing is by using only @ to change ui-view=some. This will replace the ui-view rather than checking if it is nested or not.


• Here you could use ui-sref to create a href URL dynamically on the basis of URL mentioned in a state, also you could give a state params in the json format.

For more Information Angular ui-router

For better flexibility with various nested view with states, I'd prefer you to go for ui-router