Both are templating systems, and you can use Handlebars syntax with Handlebars templating systems server-side as well.

The main difference between Handlebars-style modules and Template Toolkit-style templating is that Handlebars expects you to have all data already done as a data structure, while Template Toolkit allows you to pass in objects to the template and call methods on these objects from the template.

Personally, I've come to the point that I consider baking a data structure much better than passing objects to a template and calling methods on that object, as I prefer most logic to be written on the Perl side of the world nowadays, instead of some ad-hoc templating language with ill-defined semantics and tooling.

On the other hand, often you have tiny display logic or logic to create special links etc. where methods/callbacks into Perl are incredibly handy.