First: Take a look at this SO answer I posted, enumerating some of the differences between the two.
Now: The size of a company should have nothing to do with whether you're deploying to a Website vs. a Cloud Service. For me, I see this as a relatively straightforward decision tree:
- Do you require Linux? If so, you need a Virtual Machine.
- If you need to scale to high-end VM sizes (e.g. XL, A6, or A7), you'll need Cloud Services or Virtual Machines.
- If you need a VPN / virtual network, registry tweaks, admin-level installers (e.g. ActiveX), complex long-running installs (e.g. SQL Server), RDP/ssh access, or unsupported programming languages in Websites, you will need to go with Cloud Services or Virtual Machines.
- Regarding supported languages: Today Websites support .NET, Node.js, PHP, and Python.
If you get past the abovementioned restrictions, Websites work wonderfully. In fact, you can place your web (or web services) tier in Websites, and then put your middle tier into Cloud Services if you wanted to, and put things such as database servers and CI engines such as Jenkins into Virtual Machines. Really the breakdown is completely up to you.
Websites is really a great platform for deploying, since you get version control integration and near-instant deployments.