PHP Frameworks: Framework is a piece of software by which one can develop large enterprise, robust & secure software applications. Frameworks provide straight forward flow and strict guidelines by which developers can create any stand-alone applications with proper security and architecture. Frames itself follow some design principles. Hence developer need not to worried about application architecture. The developer needs to focus only on using a rich set of libraries and helpers to enhance application performance instead of writing the number of lines of code. Frameworks are designed given overall application security which includes database security (SQL injections), cross-browser scripting (XSS), access permissions etc. In general, the framework is an abstraction, where the developer doesn't care about how structure internally works instead of what it does. It means that the developer only needs to know how to flow rather than how flow happens internally.
Core PHP: Writing code in core PHP is right if you are developing SDK's (Software Development Kit) which acts as a third-party software for other applications. This SDK's can easily integrate with frameworks, as these are an independent piece of code. For example, payment gateways SDK's are easily integrated with other structures. Imagine if you develop SDK using framework. What will happen? Will SDK develop in CodeIgniter framework will integrate with Laravel Framework. The answer is "No". The reason is the flow and architecture of both structures are different. So it can't be integrated.
But if you are not developing SDK's it would be preferred to use the framework. It will give you the quality of code, performance, security, reusability and easy deployment.