As usual, the main advantage of a pre-built solution is that a bunch of work is done for you, and the main advantage of a homegrown solution is that you get complete control of it.
A headless CMS will do what it does, and it will come with an admin UI so that people can insert content into it without you having to build the admin UI. You will just need to set up the data structures and then pull the data out for your website.
A homegrown solution will let you structure the data however you like, easily make use of whatever existing databases you have, and do whatever server-side processing you like on the data.
A hybrid solution (where you pull data from a headless CMS with server-side code) will give you a mix of the benefits and drawbacks of each.