With server-side templates, you serve pages by loading data from the database, then produce HTML pages using templates, then ship those pages to the client. This has the following drawbacks:
Template-processing uses CPU time which can impact scaling. On the other hand, the clients often have multi-core CPUs doing nothing most of the time. We can distribute the cost of rendering pages to the clients.
Generating pages on the server increases the bandwidth necessary to serve pages, since we're sending a lot of HTML that is generated from data we load from the database. If we can send the data directly to the client and the client can generate the HTML, we can gain on bandwidth to serve the pages.
If you generate a list of items on the server using templates, it is hard to add elements to that list on the client using AJAX, because the rendering logic is on the server, which means you have to duplicate it on the client.
Add to that the fact that when doing AJAX, you always end up creating lots of HTML on the client using ad-hoc means, most of the time either DOM manipulation or using jQuery, and that leads to ugly clumsy code that is hard to maintain. This is the exact reason why the servers switched from that sort of rendering to templating long ago, which is still largely unused on the client, unfortunately.
In addition to all the benefits we already discussed, client-side templates bring a number of unique features that are not possible on server-side templates:
The ability to introduce dynamic elements in the template, such as futures (parts of the template whose rendering is delayed until a certain resource arrives from the server in the background) or timed updates (being able to re-render parts of a template at specific time intervals), or even being able to use template evaluation-time values at run-time (functions produced by the template, used in onclick handlers).
Define new template tags for new behaviour, either programatically for complex behaviour, or declaratively by simply giving a name to a template part.
Apply templates on a page automatically, or by invoking an API for dynamic updates.