What is the difference between exhaustive search and factorization in relationship to determining a key?
Your summary of exhaustive search is essentially correct; the attacker just tries every possible key until he finds one that works.  This works against almost any cryptosystem (exceptions: OTP, secret sharing, quantuum cryptography).  It's not practical against any modern system (mostly because 'has a key too large to be brute forced' is an essential attribute of any modern system).

Factorization is not a generic attack; instead, it is attack on certain public key crypto systems (mainly RSA).  With RSA, the public key contains a composite number; if someone where again to derive its factorization, they could reconstruct the private key.  That is, if someone could factor, they could break RSA.  However, a breakthough in factorization would affect only RSA (and a handful of other systems that are not used much in practice); there is no known way to use factorization to attack (say) AES, SHA-2 or Elliptic Curve Cryptography.