If you build an API for your website we call this a SPA (Single Page Application). You normally also use a frontend framework to set this up. The advantage here is that you don't need to refresh the whole page, but only parts of it. For example, if you have a table with pagination. If you would go the next page you only need to load the new rows in the table, not the whole page including the sidebar, menu, etc.
The disadvantage is that the development time mostly takes longer and can be more complicated. You have a frontend and a backend and they always need to work together.
Another advantage of an API is that you can change the frontend without changing the backend. Since they both don't depend on each other you can freely change them.
For authentication, you can use Laravel Passport or Laravel Sanctum. They are both created by Laravel and both offer secure ways for your application. It just depends on which use case you have. I would use Passport if you also have an API for the outside world available.