No matter how small what you are building, if it has any business value to the customer it will probably grow to a bigger application. As an application grows, maintaining it will become harder especially when many developers are working on it.
Frameworks have been made for many important reasons: Structure, modularity, productivity, easier maintenance, re-usable components, built in unit/integration testing, etc... I think you probably understand the benefits already.
Its not just about how fast a PHP dinosaur can churn out code, its about how maintainable the application is, and how sane and familiar the structure is. Its never about what someone 'likes', its about whats better for the project, thats what he should clearly understand.
I would suggest that you ask him about valid reasons on why he would go with such an archaic approach. Then give your valid arguments on why, his approach for this project is plain wrong. Then you can suggest some good frameworks to use. One reason behind his view might be that he is afraid of learning something like Symfony 2 because of its learning curve. You could then suggest a much simpler and straightforward framework like
Code Igniter (I wouldn't recommend it a lot but still better than nothing). Another nice framework that is a bit more complex than CI yet much more friendlier that Symfony is
Laravel 4 you should totally check it.
In this context I assume raw PHP without even using
Composer Packages and
PHPUnit. Whatever framework or approach you follow, it is absolutely a must to use a dependency / package manager. For PHP, composer is the best choice you have.
To answer your question about usage, I built things in "raw php" a couple of times in my life, that when I first learned PHP. And even then, I felt the need for structuring my application, so ended up following an MVC approach as much as possible. Then dived into Symfony 2 when it was still in beta, then Silex, Code Igniter, and now Laravel 4.
And by the way, if you have the opportunity of moving away from PHP as a language, don't hesitate. PHP is a very practical and useful language for web development, but it really lacks the fun, elegance and thoughtfulness of others like Python, Ruby or C#.
And at least, explore other languages, learn from them and try building things with them (that applies to whatever you are doing).