The two words are based on two verbs that are synonyms.
To abduct: to illegally capture someone by use of force
To kidnap: to take someone against their will; and detain them; usually for a ransom
In formal writing, especially law, use of the verb 'abduct' (The noun form is abduction.) is more common.
Although, slightly less formal, the condition of being 'kidnapped' suggests that the person can be returned if a negotiation and ransom is held for that person.
New York Times articles often use the words abduct, abduction, kidnap, kidnapper, and kidnapped almost interchangeably for the other (in their correct forms, of course). That is, they choose either noun form or verb form to mean the other.
Just a few quick examples,
(Source: New York Times)
'The security forces are engaged with the Taliban inside the city, and fighting is ... But days after the professors were "abducted" in August, the university ...'
17 Oct 2009 - NYTIMES - As they traveled to an interview with a Taliban commander outside of Kabul, .... My captors harbored many delusions about Westerners. .... our kidnapper, I still did not know which Taliban faction had abducted us.
17 Oct 2009 - Mr. Rohde, a New York Times reporter, was kidnapped with two Afghan colleagues .... My captors harbored many delusions about Westerners.