Objects and closures are two different perspectives on the same thing. That is, if you're in a language with closures but no objects, you can implement objects using closures. If you're in a language with objects but no closures, you can implement closures with objects.
The venerable master Qc Na was walking with his student, Anton. Hoping to
prompt the master into a discussion, Anton said "Master, I have heard that
objects are a very good thing - is this true?" Qc Na looked pityingly at
his student and replied, "Foolish pupil - objects are merely a poor man's
closures."
Chastised, Anton took his leave from his master and returned to his cell,
intent on studying closures. He carefully read the entire "Lambda: The
Ultimate..." series of papers and its cousins, and implemented a small
Scheme interpreter with a closure-based object system. He learned much, and
looked forward to informing his master of his progress.
On his next walk with Qc Na, Anton attempted to impress his master by
saying "Master, I have diligently studied the matter, and now understand
that objects are truly a poor man's closures." Qc Na responded by hitting
Anton with his stick, saying "When will you learn? Closures are a poor man's
object." At that moment, Anton became enlightened.
A lot of people would disagree, and hence the koan.
Personally, I see closures as objects that the compiler cooks up on your behalf. This coming from a compiler background, where said compiler is usually implemented in C++ :P.
See http://en.wikipedia.org/wiki/Closure_(computer_programming)