Azure does help in the sense that all your data is more "loss proof", and I think M$ also uses some sort of internal encryption, so some what more secure. benefits of "Azuring" your site :
1. Allowing you to expand the site, and your user base without need to jack-up your code and make major changes to your H/W, so you can grow on the fly, and all the underlying issues are taken care off, as long as you can pay up modestly( pay what you use, rather than heavy onetime investments and regular maintainance). Transitional disasters of site crashes, user loss etc will be completely avoided.
2. Azure is very well integrated into .NET; the transition will not be a major learning hurdle. Yesterday in one night I was able to figure out the blob and the table api, and how to integrate them with on of my projects.
Disadvantages:
1. Azure platform has been in flux for a lot of time, in the sense that it is not a mature product and the syntax and features are going to be in flux for sometime. ( although they have only been enhancements for the most part), so that might be an issue, for some.
2. extra effort of transition.
my suggestion: shift if you feel
1. you are expecting growth in the future
2. want to learn and experiment with new technology; and ready to make an extra effort.