Difference between mutation and DNA damage
Have you read the DNA repair article on Wikipedia?  The DNA damage and mutation section answers exactly what you're asking:


  DNA damages and mutation are fundamentally different. Damages are physical abnormalities in the DNA... [and] can be recognized by enzymes, and, thus, they can be correctly repaired if redundant information, such as the undamaged sequence in the complementary DNA strand or in a homologous chromosome, is available for copying.... In contrast to DNA damage, a mutation is a change in the base sequence of the DNA. A mutation cannot be recognized by enzymes once the base change is present in both DNA strands, and, thus, a mutation cannot be repaired.


Here's the entry from the NCBI MeSH Glossary saying the same thing:


  Injuries to DNA that introduce deviations from its normal, intact structure and which may, if left unrepaired, result in a MUTATION or a block of DNA REPLICATION... They include the introduction of illegitimate bases during replication or by deamination or other modification of bases; the loss of a base from the DNA backbone leaving an abasic site; single-strand breaks; double strand breaks; and intrastrand (PYRIMIDINE DIMERS) or interstrand crosslinking. Damage can often be repaired (DNA REPAIR).


And for mutation:


  Any detectable and heritable change in the genetic material that causes a change in the GENOTYPE and which is transmitted to daughter cells and to succeeding generations.