Differences between Azure App Services and Cloud Services
Solution 1:
Azure offers several ways to host web sites:
The same idea but in abstract vision:
Source: http://www.hanselman.com/blog/IntroducingWindowsAzureWebJobs.aspx
Solution 2:
I also had a confusion about their differences, until I read the following blog. Hope this helps.
http://cloudacademy.com/blog/microsoft-azure-app-service-virtual-machines/