Hi Scott,
I would personally prefer the Visual Studio + NuGet approach.
The Azure Gallery is great for getting up and running quickly, but its been difficult for us to get the latest version of Umbraco out in the Gallery in the last ~6 months, so for the latest recommended version NuGet is a great way to get up and running with Umbraco. And we've made a few improvements to the NuGet packages, so they should just work when deploying/building the solution to Azure Websites with git or webdeploy.
With regards to updates, its easy to handle patch upgrades, but for minor upgrades, which involves database changes, it becomes a bit trickier. The upgrade documentation covers the general approach, which should also work for the Azure Gallery. For NuGet you just update the packages, take care of config changes, deploy and then run the installer on your website to apply changes. But probably best to move these types of questions to
http://our.umbraco.org as this mailing list is intended for Umbraco Core development topics.
Hope this helps,
Morten Christensen