Both are test management tools. This document differentiate between Jira and Bugzilla

1. JIRA Vs Bugzilla
qamasterz.blogspot.in /2011/09/jira-vs-bugzilla.html
Here's a copy of an answer I found on net on comparison of JIRA and Bugz illa:
Things t hat Bugz illa does and JIRA does not :
1. Bugz illa is very good at performance with large bug databases. Take a look at bugz illa.moz illa.org, it has
~ 400,000 records. I'm not sure what hardware it runs on, but you will probably need a lot more for the same stuff on
JIRA. It's just Perl vs. J2EE. But if you have fewer than 50,000 records, don't worry.
2. Flags/requests. If you use this Bugz illa feature, you won't find anything similar in JIRA. But probably there's a plugin.
3. Authoriz ation is different. Bugz illa has a mind- breaking feature for grouping users & issues; JIRA has something
more simple (and more convenient, I think). But if you have defined a lot of security groups in Bugz illa, it may be not
easy to transfer their business logic to JIRA.
4. Search in JIRA is far less powerful than Bugz illa's advanced search.
5. If your Bugz illa is patched or integrated with other systems, take a close look at that.
6. Bugz illa is free and open- source, if that does matter.
7. Bugz illa's security theoretically should be better, because of (6).
But t here are also reasons t o move t o JIRA:
1. Web- based user interface is better.
2. JIRA supports custom fields of many types.
(LpSolit: Since Bugzilla 3.0, custom fields are also supported. In Bugzilla 3.4, it supports not less than 6 different types.)
3. JIRA (enterprise edition) supports a number of customiz able workflow schemes (LpSolit: Bugzilla 3.2 let
administrators customize the workflow.)
4. JIRA issues may be linked with custom link types of user- defined semantics.
5. JIRA has open architecture and a plug- in API and lots of plug- ins. If JIRA doesn't do something, you might find a
plug- in for that.
6. JIRA has very good commercial support.
(Someone: The commercial Bugzilla support is excellent, too)