Defect Management tool only deals with the defect tools like bugzilla, Jira..etc
But the Test management deals with all test activity start form requirement, Test scripts, Test execution, Defect management and tools like Quality center, Test Director.
Defect management is also part of test management.
EX: QC , Iplan( Used in EMirates airlines ) are the testmanagement tool .
which includes Requirements ( SRS , SDD doc as per sprint wide ) , Releases ( phase wide doc mapped from req)
Testplan ( includes the testcases maintainance ) , Testlab ( includes the testcase execution ) , Defects ( Defect / CR /spec issue maintainance ) and This last tab called Defect management tool )