Full disclosure from me as well: I have been working at Contentful for the last few years, before that and for at least 10 years, however, I have been building sites using a wide range of CMS. Sitecore, EPiServer, Drupal, Wordpress, Umbraco, Magnolia etc.
The main difference, that is very important to understand, is that a truly headless CMS comes completely decoupled from the presentation of content. There are no templates, there is no hierarchical view of content that ties to the presentation (e.g. a page tree) and there is no need to host the actual CMS anywhere.
This leaves you free to "only" develop the application while the CMS simply gets out of your way. No tech stack is imposed on you. No assemblies need to be scanned to get your models. No local database needs to be set up for development.
The downside is that you need to do some more work to get the actual data from a headless CMS since for every piece of content you need, you make an API call. This is normally handled through officially supported SDKs (for example https://github.com/contentful/contentful.net) for whatever language you want to fetch the content for.
On a feature to feature comparison, the headless CMSs often fall short of the more traditional CMS vendors, but the key to understanding here is how a headless CMS encourages a true microservices architecture where you use the right service for the right task, instead of relying on a monolithic "solve them all" platform. By integrating a headless CMS with personalization from, for example, Optimizely or Google optimize, you get the benefits of each platform and can use their respective strengths to improve your final solution.
Here's a blogpost I wrote the first time I stumbled upon Contentful (before I started working there) https://robertlinde.se/contentful/2016/04/18/contentful-the-future-of-cms.html that might give you a few more ideas of the benefits.