To my way of thinking, the primary difference is in the "open" approach vs. the "integrated stack" approach. Microsoft likes to provide everything as an integrated stack that all shares a common flavor and approach. Java is more friendly to the "bring your own x" model, where you may want to plug in your favorite application server, transaction manager, etc. Both technology stacks allow in-process invocation as well as remote invocation with varying levels of transaction support.
Really, WCF is not a new technology stack, but a reorganization and rebranding of existing elements of the .NET stack. Specifically, WCF took on the functions of .NET Remoting, Web Services, and distributed transactions.
You reference "the more common current approach which is just to have business objects as part of the web server's ASP.NET process." That is only common for non-distributed apps. It is a simple model that works well when all of your objects will reside on the same server. This follows Microsoft's "Scale Out" deployment model. Rather than segregating object tiers across servers, put everything but the database together on the web servers and then incrementally add identical, redundant servers to scale out the web-server layer.
Microsoft has been pushing hard lately on Service Oriented Architecture, which relies more heavily on WCF and remote invocation. This is seen as a key to the cloud strategy that would have people moving parts or all of their applications to flexible resources in the cloud (which MS would like to host with Azure and the like).