On any multi-person project, the number of details to track and bugs to fix grows exponentially. Trying to remember everything yourself (or on Post-it Notes stuck to your monitor) will result in some important issue slipping through the cracks.
A defect tracking system is a database that stores all your bug reports (Problem Reports, or PRs) in a single place. The system becomes the authoritative source of data about the status of your project. Even if someone leaves the team, much of his knowledge about open bugs and how past bugs were fixed will be retained in the system.
Even if you don't realize it, your organization has a workflow: Each person finishes his share of a task and sends the result to someone else, much like an assembly line. Mistakes can cause workflows to skip over important steps. But defect tracking systems can keep track of each issue's state, ensuring the appropriate owner is aware of all her issues.
Very often, a developer fixes a bug but then the tester discovers it isn't really fixed. This is called a bounce-back. Make sure your workflow expects these and handles them by retesting every bug fix. Preferably, the person who created the bug will be the one to acknowledge whether the fix works.
Storing all PRs in one place facilitates searching and reporting. Want to know which programmer currently has the most open PRs, or whether bugs are being reported faster than they're being fixed? Want to know what percentage of PRs are bounced back to development, or whether a similar bug has ever occurred before? A good tracking system can tell you.
People other than developers need access to the PRs. Tech support, documentation writers, testers, and even product management can all benefit greatly by searching through the defect tracking system.