BUGZILLA : 

• It is an Open SOURCE
• Using Bugzilla might be little complicated for few due to grouping users and granting permissions
• Bugzilla allows you to show/hide the whole custom field or specific values based on the VALUE of some other field
• Bugzilla’s has a powerful advanced search option
• Unlike JIRA, Bugzilla allows users select the initial status of a new issue.
• Bugzilla has only one link type: Blocks/depends and a Bug ID custom field

JIRA :

• It is a commercial tool.
• For some using JIRA would be more convenient than Bugzilla
• JIRA ENABLES conditional configuration based only on Type fields and Project.
• JIRA lacks advance-level search options. JIRA has flexible JQL language (JIRA Query Language). It enables you to build ARBITRARY boolean expressions.
• Unlike Bugzilla, JIRA enables you to define multiple workflows which are applied based on the issue’s Project and Type.
• JIRA has configurable link types with user-defined semantics. JIRA enables to link an issue to any other entity outside JIRA.

Bugzilla : 

JIRA :