We want to build a Help Desk issue tool in our SharePoint site. Should I use the Task template or the Issue Tracking Template?

This depends on the functionality you need.

If you are looking for the ability to enter tickets, categorize them, notify people when tickets have been assigned, then the Issue Tracking template is excellent.

But, if you wish to have your tool sync with MS Project or Outlook, you would need to use a Task template. The task template also gives you more views and a timeline.

So, keep your needs clear. Are you tracking an IT project or Help Desk tickets?