1) There is a common misconception that traditional taxonomy is based on morphology, while cladistics is based on genetic data. How is this misconception incorrect?
(2) Because biologists disagree about the details of phylogenies, some nonscientists argue that such disagreements warrant the questioning of evolutionary theory in its entirety. Why is this reasoning flawed?
Traditional taxonomy is the grouping and naming or organisms. While it traditionally was based on morphology (as that is basically all Linnaeus had to go on without modern technology), taxonomy is now predominantly based on other characteristics such as similarities between DNA and RNA, therefore taxonomy is now also based on genetic data.
Cladistics is the study of organisms based on the time in which different groups arose (or became a distinct group) relative to one another which is mostly based on genetic information. It is a relatively newer branch of phylogeny compared to traditional taxonomy.
Taxonomy and cladistics are not necessarily opposing ways to classify organisms since one primarily deals with grouping and relationships with organisms and the other tries to put these groupings and relationships in chronological order. Therefore, bothe of these are needed to make a proper phylogenetic tree as they cover slightly different areas. For example, taxonomy might tell us that marsupial mammals are closer related to each other than to other vertebrates while cladistics may tell us how long ago it’s been since marsupials and other vertebrates shared a common ancestor (therefore taxonomy may tell us what that common ancestor is but cladistics tell us when that was).