Live re-rendering of data -- imagine if the (now) "24 minutes ago" timestamp next to your handle was always accurate, instead of just reflecting when I loaded the page.
Easier implementation of otherwise difficult features. Doing a client-side autocomplete of people's names is trivial when you already have all of the other accounts in your organization modeled in JavaScript ... but not as fun when you have to ajax for server-rendered HTML for it.
Live updates of pushed data. When another reader +1's a post that you're currently looking at, if the server pushes that data to you, and you have a model for that post -- it's easy to increment the counter. If the server has to push the re-rendered HTML for the entire post, it's much more difficult.
... and those are just the tip of the iceberg.