I've recently done work at a company where there are no automated tests, and tests aren't part of the culture. I've been writing tests for so long now that it's jarring to consider working without them, so I've gone ahead and written tests for my part of the work. As I've done so it's really brought home the advantages of writing tests, and I wanted to list them out.
Tests Make Your Code Work
The earlier you write a test for a piece of code, the sooner you'll find yourself stepping through it, double-checking that and how it works. Working with code through tests is a unique experience, and things occur to you when you're stepping through a class that won't when you're reading through it. What if that variable turned out to have that value? What if that resource turned out to be unavailable? Watching your code go through its paces brings home the fact that it's going to be part of a functioning system with real users one day, and makes you consider if it does its job properly.
Tests Give You Easy Entry Points To Diagnose Problems
If something goes wrong in a particular part of your system, tests give you a way to access that part without going through the various layers that usually sit between it and the outside world. Without tests you have to crank up the entire system and work your way from the user interface to the situation that caused the error. That could include getting a user into a particular state or entering a particular combination of data - do any of that even a little bit wrong and you can easily find yourself no closer to a solution, proclaiming that "it works on my machine". Worse still is if the error was caused by something transient, or a quirk in a third party system only triggered by certain circumstances - with tests you can easily stub out components and reliably simulate that behaviour so you can check how your system deals with it. Without tests... well... good luck with that.
Tests Guard Against Other People's Changes
A colleague working without tests reported the other day that he'd spent most of a day trying to figure out why part of the system he'd written had stopped working. He'd been sending the same data to a third-party service all along, and up until that morning it'd always worked; he now wasn't sure if something had changed with his setup or what he was doing, or something had changed with the third party. Tests would have stopped this problem before it got started - he would have had a failing test to go to telling him what had changed.
Tests Are Specification
You have a specification which says in circumstance ABC the system should perform action XYZ - does it? Sure, you stepped through it those times and saw it do XYZ, but a lot has happened since then - does it do XYZ now? Tests are an executable specification of what your system does. Not what it's supposed to do, but what it does. Tests are the only specification you have which the code definitely fulfils. You probably have a Word document which says how your system is supposed to behave, but your tests tell you how it actually behaves. In every area of life, you only know what you can demonstrate - tests are a great way of demonstrating what your system does.
Tests Are Documentation
How are the classes and subsystems in your system used? What inputs do they expect? What outputs do they give? Do they have any quirky behaviour? Do they throw exceptions or returns nulls or empty collections? That third-party library you're using to validate bank account details - does it say account number 111111 with a valid sort code is valid, or invalid? You can write comments, Sandcastle comments, Word documents and make UML diagrams all you want, but your tests are the only documentation you can rely on to be accurate. If code changes, passing tests have to change with it - comments, documents and diagrams don't.
Tests Are Insurance
You put an application live and move on. A few months later you revisit it to add a new feature - without tests, how easy is it to tell if your new feature breaks any of your old features? This is a few months down the road and your familiarity with the existing system has faded, so your intuition isn't going to help you as much as it once did. This is another area where tests pay huge dividends - if your new feature breaks an existing one, your tests will let you know. I've added to and refactored code on a well-tested project where I've genuinely had no idea (and put next to no effort into figuring out) if it would break something - safely insured that if it does, my tests will tell me. Without tests you're stepping on eggshells whenever you need to change existing code.
...and that's why tests are so great. I've been working with them for so long now that it baffles me how anyone writes anything non-trivial without them. Getting used to writing tests takes some discipline and effort, but it's repaid many times over - if you don't write tests, you owe it to yourself and your team to start.