Could somebody explain this difference just as simple as possible?
Pattern recognition is the “automated discovery of patterns in a training set”, and so it is a general term for machine learning.
Classification is the supervised learning problem whose target value is a finite set of classes (as opposed to regression, wherein the target value is a continuous variable).
Therefore, classification is a kind of pattern recognition problem.
Classification is simply a more general term than pattern recognition. In both cases, you have a set of classes $K$ and a collection of observations, where each observation is represented by a set of features. Your job is to find a mapping of features to a member of $K$ such that you minimize some measure/estimate of out-of-sample classification error.
In pattern recognition, you are simply using a very complex/large feature space. For example, facial recognition will have an input space equal to the number of pixels in the image. This is no different than classifying a loan application as high or low risk based on several measures of creditworthiness.