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 )
Defect Management Tool is used to track the defects only.
Test Management Tool is used to plan tests, track requirements, test cases development, monitoring requirements coverage, test cases execution (manually), defect tracking and reporting.
Iam not sure whether Iam correct but according to me Test management Tool is used to for tracking Requirement , making test plan and also execution test plan and bug reporting.
Defect management is tool to track the severity of bugs in each Phase wherein the project manger address all the severity of issues and it status in tool.