It's complex at first but there's very little controversy to be found.
Noam Chomsky is a linguist and political activist famous for revolutionizing the study of all areas of linguistics via structuralist methods. He also has a second life as a political critic, bringing scientific methods to journalism, measuring coverage of topics to show the bias of news organizations.
Tom Wolfe is an author famous for the New Journalism movement in the 60's, writing a number of well-received novelistic non-fiction books about American culture. His most famous book is The Right Stuff about the American space program.
Wolfe recently wrote a book, The Kingdom of Speech critiquing two things, Darwin's theories of evolution and Chomsky's theory of Universal Grammar. To oversimplify considerably, "the ability to learn grammar is hardwired into the [human] brain". It is an overarching abstract theory, which also has a lot of very particular falsifiable details. Although there are academic alternatives to the big picture and to the details, it sets the program to explain the commonalities of human language (especially grammar). And there are controversies over some of the details e.g. possible lack of recursion (Everett/Pirahã), unfalsifiability (there are universals to language by definition of a word 'language' (but then there certainly are language specific areas of the brain)).
Wolfe's critique seems to boil down to 1) the single Pirahã questionable counterexample, 2) "language is not the result of evolution but essentially a verbal trick that was invented by human beings", and 3) "It's a memory aid – a mnemonic – that enables human beings to store away a piece of information and compare it to a new piece of information and draw conclusions." The first is a reasonable scientific critique, but has been judged by linguists not to be sufficient. The second is a broad nature vs nurture claim, which is an understandable but false dichotomy to make; it is obvious that there are components to be learned and also that there are parts where biology is necessary, and much interaction between. The third seems to treat language as simple vocabulary, and as such really hardly addresses any of the concerns actual scientists have about UG.
Wolfe also seems to compare externalities: theoretical linguists tend to pasty-skinned sunless armchair theorizing, Everett is the dashing revolutionary storming the barricades of the ossified ivory tower.
Also, there's no actual dispute between Wolfe and Chomsky, in the sense that there's little to no dialog between them. Wolfe has criticized some implications of Chomsky's theories and Chomsky has only made comments about the book saying the "errors are so extraordinary it would take an essay to review them".
In short, Wolfe makes for very entertaining intellectual reading but is not a good or even relevant source of the substance of the UG controversy.