Web services are great if you are only using simple data transfers between the client and host. You form up the XML message, send it to the server and get a simple XML message, back. You then deserialise it and act upon the data accordingly. AS3 support for XML is superb, you can use it to communicate with any data service and you are in full control at the client end.
Where web services fall down is when the data structures passed back and forth become too complex, or when the number of different data structures passed back and forth become too large. Remoting overcomes these problems by providing a heavyweight framework that handles the serialisation for you. Define your .NET/ PHP/ Java or whatever classes, and the framework should provide tools to generate the equivalent AS classes. That way you can send complex object structures back and forth without needing to know how the data is being serialised. Because you don't need to know, it can compress the structure to make it pretty much non-human-readable, or even use binary data so you get speed increases too.
Remoting is not suitable for small scale or varied server protocol stuff though. You have to synchronise class structures between server and client; you can only communicate with a compatible remoting servers and the framework adds overhead to the client size and complexity.
There is no right answer as to which to use, when. Both have advantages and disadvantages. As a rule of thumb though, use web services for simple stuff and remoting for complex stuff (which is pretty vague advice ;)