Does the op-amp output depend on the difference in input?
An op-amp does amplify the difference between the two inputs, but the gain of an op amp is generally very, very high (>10,000) so the input difference required to drive the op amp output from one end of the range to the other is very small.  For an op amp with a gain of 10,000 on +/- 15 volt rails, the output swing is 30 volts while the input difference to drive that output swing is 3 mV.  In other words, the difference between the + and - inputs of this op amp should not exceed 3 mV so long as the output is not being driven into the rail.  An ideal op amp simply takes the limit as the gain goes to infinity, driving the input voltage difference to zero.  

This is done because the gain of the op amp is not necessarily linear - it can vary depending on the input and output voltages.  Generally the gain will decrease as the output voltage nears the rails.  The gain can also depend on the common-mode input voltage as well.  The high gain means that when you add a feedback loop, the closed-loop gain will be much more linear than the open-loop gain, even though the op amp open-loop gain is not so nice.