The Merriam-Webster definition of "classification" is "systematic arrangement in groups or categories according to established criteria." The term is a broad one that encompasses any type of grouping according to criteria. For example, placing multicolored candies in groups according to color is classification. Humans naturally classify things because groups of things are easier to understand than a multitude of unrelated things. Classification usually does not extend beyond one criterion or two criteria.
The Merriam-Webster definition of "taxonomy" is "orderly classification of plants and animals according to their presumed natural relationships." Taxonomy is the process of giving names to things or groups of things according to their positions in a hierarchy. For example, the taxonomy of the animal kingdom organizes animals into smaller and smaller groups, with each group being a subset of the groups above it. The items are defined according to their relationship with the other items in the hierarchy.
Classification and taxonomy are both methods for organizing and categorizing large amounts of data in a form that humans are able to comprehend. They are tools that allow us to maintain databases of separate but related items so that those items can be easily compared and contrasted. They describe the items in a way that makes it easy for us to return to them later without having to analyze each piece of data every time we need to use it.
Taxonomies are more concerned with providing exhaustive lists while classification is not exhaustive. Taxonomies are based on providing a hierarchical relationship map between a multitude of items while classification usually only groups items according to one or two attributes. The fundamental difference is that taxonomies describes relationships between items while classification simply groups the items.