Are the words "wrapper" and "shim" more or less synonymous, or is there a distinction, perhaps connotative?
A shim is typically something written specifically to maintain backwards compatibility. For example, if you have two versions of an API, version 1 and version 2, then rather than maintaining version 1 independently of version 2, you might write a shim that intercepts calls to version 1 of the API, translates the parameters to what version 2 requires, and then returns the results.
Typically, a shim is written by the provider of the API, rather than consumer.
A wrapper is written by the consumer of an API and is typically written so that you can switch the underlying API without the rest of your code having to know. For example, you might write a database wrapper so that you can talk to MS SQL Server, or Oracle, just by switching wrappers.
Of course, as with any terminology, there are gray areas, but I think the above covers the main differences.