Long story short - I've a very pleasant experience while trying Youtrack now. But let's take a closer look!
The command-line with auto-complete. Sometimes I get really lazy and just unwilling to use the bugtracker as I have to use the mouse and click over several screens in the bugtracker's UI to get the job done. With Youtrack, I can make queries in the command-line that has autocompletion, which is awesome!
Project settings and administration. It took me just a couple of minutes to configure a project and the users. And also I figured out very quickly how to the user roles work. Considering I'm just an ordinary user for this type of software - this is a good sign!
Shortcuts are configured just like in IntelliJ IDEA, which I'm being a big fan of. I think this is an awesome feature for any IntelliJ user trying to use Youtrack.
The UI did not cause any animosity during use for me. Also, you can make screenshots just from the form where you report an issue - a very handy feature I have to say.
Talking about the price. It might seem from the first sight that the price is not as friendly as the UI. But actually if I just compare it to Atlassian JIRA, then it is not as bad maybe. I know it is not correct to compare Youtrack to JIRA feature wise. JIRA is a full-blown issue reporting system with some project management facilities; lots of plug-ins, lots of customizations, etc. But if I keep in mind that my aim is just to find a cool software that could be used as a public bug tracking tool for some project, I keep the features comparison aside for now. At the end, the unlimited seat license for JIRA is approx. 5 times as expensive as analogical Youtrack license.
One may ask, why pay if there are tools like Bugzilla, Mantis, etc. I think that Youtrack's command-line with autocompletion partially answers this. There're some more nice features that you may find, but the general reason is the usability of this tool. Especially, Youtrack might be a good choice if the company is already using some other products from JetBrains, e.g. IntelliJ IDEA or TeamCity.
What is missing? I've heard that some people miss the workflow feature. But just recently I discovered that this is a work in progress and custom workflow will be available very soon. Also, myself, I'm missing the hosted mode for Youtrack, and the integration with some version control system (just like FogBugz integrates with Kiln). AFAIK, the hosted Youtrack version will be available soon, so I'll be just waiting for a good integration with Mercurial or Git repositories :)
In general, I think Youtrak is a very good tool to be used as a public bug tracker software, where a user can report an issue and see if the issue was taken in works, was it assigned to a developer, and was it fixed after a new release.