Building an application + JSON API in Laravel

Namespace everything. You can keep it all inside the same application then. There is no point in maintaining 2 code bases because you will have to repeat your business logic in 2 places.

In your routes, then you can then do this

Route::controller('user', 'UserController');

Route::group(['prefix' => 'api', 'namespace' => 'Api'], function() {
    Route::controller('user', 'Api\UserController');
});

Also, don't write your business logic in your controllers. Use commands (known as jobs in Laravel 5.1) and repositories instead.

Pretend you have a Create User feature. Then you will have a corresponding Command/Job class for that.

namespace App\Jobs;

use App\Repositories\UserRepository;
use App\Jobs\Job;
use Illuminate\Contracts\Mail\Mailer;
use Illuminate\Queue\SerializesModels;
use Illuminate\Queue\InteractsWithQueue;
use Illuminate\Contracts\Bus\SelfHandling;
use Illuminate\Contracts\Queue\ShouldQueue;

class CreateUser extends Job implements SelfHandling, ShouldQueue
{
    use InteractsWithQueue, SerializesModels;

    protected $user;

    public function __construct(UserRepository $user)
    {
        $this->user = $user;
    }

    public function handle(Mailer $mailer)
    {
        // logic to create user
    }
}

Which you will use in your UserController

public function postCreateUser()
{
    // validate request

    $this->dispatch(new CreateUser($inputData));

    // return view
}

and then your Api\UserController

public function postCreateUser()
{
    // validate request

    $this->dispatch(new CreateUser($inputData));

    // return JSON output
}

I would chose separation. I like to approach like this:

//routes.php
Route::group([], function()
{
    Route::match(['get', 'post'],'/', ['as' => 'homepage', 'uses' => 'SiteController@index']);
    //other frontend routes
});

Route::group(['namespace' => 'Admin'], function()
{
    Route::match(['get', 'post'],'/admin', ['as' => 'admin', 'uses' => 'SiteController@index']);
    //other admin/backend routes
});

Route::group(['namespace' => 'Rest'], function()
{
    Route::match(['get', 'post'],'/rest', ['as' => 'rest', 'uses' => 'RestController@index']);
    //other rest routes
});

Controllers for frontend in Controllers folder. Controllers for admin in Controllers/Admin folder. Controllers for rest in Controllers/Rest folder.