Specter initially covered local news at The Washington Post in 1985, then became a national science reporter for the Post and finally New York City bureau chief. In 1991, Specter transferred to the Times, where from 1994 to 1998 he was based in Moscow. In 1995, he was appointed co-chief of the Times Moscow bureau, and while in Russia he covered the war in Chechnya, the 1996 Russian presidential elections, and the declining state of Russian health care among other stories. In 1998, he became a roving correspondent based in Rome covering topics as varied as Europe's demographic crisis, Michelangelo's Florentine Pietà and the spread of AIDS in Africa.

His 2009 book, How Irrational Thinking Hinders Scientific Progress, Harms the Planet, and Threatens Our Lives (ISBN 978-1594202308), explores the ways in which people in the United States and Europe have increasingly rejected scientific truths, backed by impressive data, instead embracing what often seem to be more comfortable fictions — about issues such as the value of organic food, vaccine safety, and personal genomics. He recently delivered a talk titled "The danger of science denial" at TED 2010.

At The New Yorker, he has written about the global AIDS epidemic, avian influenza, malaria, scientific efforts to resurrect extinct viruses, synthetic biology, genetically modified food, efforts to mine the human genome to fight disease, and the world’s diminishing freshwater resources. He has also written profiles of many people, including Lance Armstrong, Richard Branson, the ethicist Peter Singer, P. Diddy, Manolo Blahnik, AIDS activist Larry Kramer, and Ingrid Newkirk, the highly controversial founder of PETA.

In 1996, Specter was awarded the Overseas Press Club Citation for Excellence for his coverage of the War in Chechnya. In 2002, he won the A.A.A.S. Science Journalism Award. He has also twice received the Global Health Council's Annual Excellence in Media Award- for his piece about AIDS in India, “India's Plague” (12/17/01) and for one about AIDS and the population crisis in Russia ,“The Devastation,“.

In 2009, Specter received the Robert P. Balles Annual Prize in Critical Thinking for his book Denialism. The yearly award is given by the Committee for Skeptical Inquiry to the author of the published work that best exemplifies healthy skepticism, logical analysis, or empirical science.

Specter is a son of Howard and Eileen Specter. He was previously married to Alessandra Stanley, now a television critic for The New York Times. They have one daughter, Emma.

Specter is a 1977 graduate of Vassar College.