Both Cloud Services and Windows Azure Web Sites are fine hosts for an ASP.NET Web API.
I agree with @SLaks - depends on how much control you want.
Use Cloud Service for maximum control (including control over performance characteristics and (for APIs that need this) ability to scale out). Web Sites for speed of deployment (seconds rather than minutes) and cost (might be free, depending on your scaling needs).
- With a Web Site, your largest VM size is Large. With a Cloud Service, it can go to XL (twice the specs of a Large).
- With a Web Site, you can have up to three VM instances in Reserved Mode. With a Cloud Service, there is no hard limit (though you may need to call support to exceed 20 cores in a standard account).
- With a Web Site, you have no control over IIS. With a Cloud Service, you do, so you can do things like use the Warm-up Module in IIS and tweak the recycle settings (these both help improve latency in certain scenarios).
More on Scaling Web Sites: http://www.windowsazure.com/en-us/manage/services/web-sites/how-to-scale-websites/
In summary, from least-to-most control, you really have: Web Sites Shared Instances (high density mode, little control, limited scale, free) => Reserved Mode (little control, better scale, more predictable (b/c no tenants you don't know about), costs money, low knowledge) => Cloud Service (max control over IIS, costs money, more knowledge needed around Cloud Services).