It's another way of running rich functionality in a browser. It might be advantageous in some situations to solve uncommon difficulties with distributing UIs. It won't necessarily be a tool for the common case.
Silverlight and Azure are different things, but they're both marketed towards developers who would like to take functionality, much of it in the form of internal apps, and make it accessible on the web or otherwise over WAN. I know this, because some Microsoft folks were pitching Azure to me just yesterday!
Yes, things don't make sense if you're stuck on "compete = comparable." A little reflection, and you should realize this isn't a very good assumption. (Teleconferencing and airlines, anyone?)