Why are New HubSpot CMS Themes Better for Websites Than Old HubSpot Templates?

The new HubSpot Themes are supported by HubL templates (new) vs. HubSpot, Drag and Drop (old) - it pays to know the difference

In terms of the technical perspective, I only see positives of using the CMS functionality in its most updated version. As the customer mentioned, there is far more flexibility when using HTML + HubL templates.  Here is the article we’ve been referring users to help push them towards themes. It is also worth noting we are on a slow path to deprecate the legacy layout templates, you can see the announcements we’ve made for templates in the marketplace here and here. If they are looking for more technical resources I suggest they check out our developer documentation. It is worth noting that most of the developer documentation focuses on themes and coded templates which serves as another reason as to why investing in a site that is built with the updated functionality would most likely be the best investment. Themes are compatible with all of the latest and greatest content editing features that will enable your customer to create content much more quickly and easily and are actually much more flexible for both end-users and developers than legacy templates.
To touch on their question regarding tracking and sales mechanisms, In the next few years, we will continue to strengthen the relationship between our CRM and CMS. Themes and coded template files will play a pivotal role in this as a lot of this functionality will be packaged within a theme such as data-driven content and commerce. From a sales perspective, all new custom quoting capabilities are built on themes and we will continue to grow a suite of sales content types over the next few years. Ultimately if they don’t choose themes, they will shut themselves off from all the amazing advancements we continue to make. 

To be transparent, I do not know in what regard the other agency holds Templates (DND format) over Themes. I have not heard this across any of the Partners that I manage. It may be helpful for the customer to understand the reasons behind this recommendation as a first step. If they can share the argument that the other agency is making for the "older" templates, then perhaps we can do a better job of combating those specifically. 
1) your quote "As the customer mentioned, there is far more flexibility when using HTML + HubL templates." is this what you mean, that they have more flexibility using templates rather than themes? Does this go against what we are saying -- to move to themes? 
2) Your very first link your mention this is what we are using to push people to themes - I don't really see anything that talks about the why
In regards to the clarification question, let's take a step back. Themes are a set of files that include coded "Templates" among other files mentioned in the resources I sent over. Since templates are still a large part of themes, it is important to understand the difference lies in the functionality available within the template in question. To explain further, there are essentially two "types" of templates when it comes to HubSpot, Drag and Drop (old) vs HTML + HubL templates (new). I highly recommend gaining familiarity with the two as this may come up more often than not. One thing to note: it may be worth asking what type of template is being referenced in future CMS-related conversations when clarity is needed.