I've read a lot of summaries of block ciphers particularly with regards to the NIST competitions stating that reduced-round block ciphers are, for example, vulnerable to differential cryptanalysis.
I have a general idea that the application of differential cryptanalysis is to look at the difference between inputs; Wikipedia makes that fairly clear. However, I could take any two inputs for any given block cipher and I am pretty certain I'd be staring at random differences.
I am aware this is the idea of a well written block cipher; however, assuming a broken or vulnerable cipher (feel free to choose simple examples) how do I go about choosing differences to try?
Are there any clues in algorithm design that would inform a decision on which values to choose?
How does being vulnerable to differential cryptanalysis impact a cipher in the wild?
If all I have are differences between known plain-texts and known keys as my analysis and a captured ciphertext as my data to exploit, what can I actually deduce?