The simple difference between Linnean systematics and cladistic classification is that they represent the two possible kinds of classification: consistent and inconsistent, respectively. The Linnean system of classification is a consistent classification, whereas cladistic classification is an inconsistent classification. Naive classification like cladistic classification is ultimately paradoxically contradictory, as Bertrand Russell demonstrated in 1901, which the Linnean system of classification bypasses by its orthogonal structure.
This simple difference is not immediately obvious if one confuses classification with dichotomous branching processes, as “cladistics” does (and originally the Nazi race biologist Willi Hennig did), but the problem with thus confusion is that neither such processes can be classified consistently using naive classification. The difference may thus not be immediately obvious, but it’s none the less a fact.
The fact that the difference is not immediately obvious, in turn, unfortunately means that there will be people that deny it and people that can’t understand it, but who instead sets out to find the confusion of the difference. These people are today called “cladists”.
“Cladistics” is thus in practice a melt-down of the foundation for science, ie, classification, thereby confusing everything that possibly can be confused.