Can someone illustrate what really is the difference between the two?
This question has been answered a few different ways, though I think a concrete example is missing.
Design Pattern: A set of rules put together to solve a reoccurring problem. The rules define the pattern. There is no code you can copy paste to implement this. The developer is supposed to understand the rules then implement them.
Example:
Design Patterns are split into 3 categories:
I've found this link to be helpful in explaining them.
Framework: A framework, not to be confused with libraries, is the basic structure underlying a system that allows developers to build and deploy software applications in a standardized way.
Example: The .Net Core framework is a tool that contains a set of libraries that enables developers using the Microsoft stack to write software applications that can be deployed in multiple different environments (Windows, Mac OS, Linux)
Library: A reusable set of methods/classes put together in one package to provide functionality.
Example: The System.Collections.dll library enables .Net developers to use things such as a Hashtable.
It's worth it to also look at other answers, as sometimes it's just a buzzword and it differs from one environment to another. Look here.
I hope this helps.