Although I take the risk of not being popular I say they are not useful nowadays.
I think they were well intended and useful in the past when for example DELETE told the server to delete the resource found at supplied URL and PUT (with its sibling PATCH) told the server to do update in an idempotent manner.
Things evolved and URLs became virtual (see url rewriting for example) making resources lose their initial meaning of real folder/subforder/file and so, CRUD action verbs covered by HTTP protocol methods (GET, POST, PUT/PATCH, DELETE) lost track.
Let's take an example:
On the left side is not written the HTTP method, essentially it doesn't matter (POST and GET are enough) and on the right side appropriate HTTP methods are used.
Right side looks elegant, clean and professional. Imagine now you have to maintain a code that's been using the elegant API and you have to search where deletion call is done. You'll search for "api/entity" and among results you'll have to see which one is doing DELETE. Or even worse, you have a junior programmer which by mistake switched PUT with DELETE and as URL is the same shit happened.
In my opinion putting the action verb in the URL has advantages over using the appropriate HTTP method for that action even if it's not so elegant. If you want to see where delete call is made you just have to search for "api/entity/delete" and you'll find it straight away.
Building an API without the whole HTTP array of methods makes it easier to be consumed and maintained afterwards