Bugzilla is bigger, a larger community, more features, more power ... for that reason I've always prefered mantis ;) Mantis is as ugly as sin but for most projects it gives you what you need in a simple and intuative way.
If you have a large team, a big QA department and all the rest bugzillia may be a better fit. Small team that just needs to get stuff done - then mantis is probably better in my opinion.
The biggest feature missing from mantis (they may have added it since, this was a few years ago) is the reports feature so you can track progress with nice line and pie charts. However, I just wrote a simple PHP script to pull out the data and manually created them in Excel each week (only took 5 minutes or so). Not great but functionally sufficient for what we needed at the time.
However there online demos of both so I suggest you try them out and pick what suits you the best.
