I'm a huge fan of Selenium. Saying 'unit-testing your web ui' isn't exactly accurate as some of the comments have mentioned. However, I do find Selenium to be incredibly useful for performing those sort of acceptance and sanity tests on the UI.
A good way to get started is using Selenium IDE as part of your development. Ie, just have the IDE open as you're developing and write your test as you go to cut down on your dev time. (Instead of having to manually go through the UI to get to the point where you can test whatever you're working on, just hit a button and Selenium IDE will take care of that for you. It's a terrific time-saver!)
Most of my major use case scenarios have Selenium RC tests to back them up. You can't really think of them as unit-tests along the lines of an xUnit framework, but they are tests targetted to very specific functionality. They're quick to write (especially if you implement common methods for things like logging in or setting up your test cases), quick to run, and provide a very tight feedback loop. In those senses Selenium RC tests are very similar to unit-tests.
I think, like anything else, if you put the effort into properly learning a test tool (eg, Selenium), your effort will pay off in spades. You mention that your company already uses Selenium to do UI testing. This is great. Work with it. If you find Selenium hard to use, or confusing, stick with it. The learning curve really isn't all that steep once you learn the API a little bit.
If I'm working on a web app, its rare for me to write a significant amount of code without Selenium RC tests to back it up. That's how effective I find Selenium. :) (Hopefully that'll answer your question..)
