While the dictionary has the words as near-synonyms, taxonomies are the result of systematics. Systematics is a technique or procedure that results in a taxonomy. Linnaeus’ taxonomy of plants is the result of his applying systematics to the problem (in this case,applying differences in leaf structures, reproductive means, etc.); Mendeleev’s chart of the elements was the result of his applying systematics to the problem (in his case, number of protons, electrons, etc.).
If you were given the task of organizing the miscellaneous drawer in your kitchen, you would formulate some logical pragmatic method for dividing the items into a taxonomy. For example, the uses of the items (fresh batteries, candles, matches, would be put into the category of “things needed if the electricity goes out) or playing cards, dominoes, chess set (“games”) or “paper clips, rubber clips, clamps, twist-ties” (fasteners), etc.—The logical procedure that breaks things into categories is “systematizing.”
When philosophers say “Taxonomies are never innocent”, they are referring to the criteria used in the systematizing procedure. Organizing toys for a garage sale, for example, would use a different set of systematizing criteria than organizing the toys in a boy’s bedroom. So, systematics is the study of the methodology; taxonomies are the product of applying those methods to a specific sorting-out task.