First of all, like many answers state here: classification is supervised learning and clustering is unsupervised. This means:
Classification needs labeled data so the classifiers can be trained on this data, and after that start classifying new unseen data based on what he knows. Unsupervised learning like clustering does not uses labeled data, and what it actually does is to discover intrinsic structures in the data like groups.
Another difference between both techniques (related to the previous one), is the fact that classification is a form of discrete regression problem where the output is a categorical dependent variable. Whereas clustering's output yields a set of subsets called groups. The way to evaluate these two models is also different for the same reason: in classification you often have to check for the precision and recall, things like overfitting and underfitting, etc. Those things will tell you how good is the model. But in clustering you usually need the vision of and expert to interpret what you find, because you don't know what type of structure you have (type of group or cluster). That's why clustering belongs to exploratory data analysis.
Finally, i would say that applications are the main difference between both. Classification as the word says, is used to discriminate instances that belong to a class or another, for example a man or a woman, a cat or a dog, etc. Clustering is often used in the diagnosis of medical illness, discovery of patterns, etc.
