OK, I think I have found the answer to my own question. (It was buried under the tabs on the SS Express product page.)
Here is a summary of the distinctives of LocalDB I found on that page:
- LocalDB is a lightweight deployment option for SQL Server Express Edition with fewer prerequisites and quicker installation.
- LocalDB has all of the same programmability features as SQL Express, but runs in "user mode"* with applications and not as a service.
- LocalDB is not intended for multi-user scenarios or to be used as a server. (If you need to accept remote connections or administer remotely, you need SQL Server Express.)
- "Express with Tools" (which includes SS Management Studio Express, Azure, etc) can be used with LocalDB or without. (The same goes for "Express with Advanced Services".)
UPDATE: I just found this useful description in Windows IT Pro (Jul '12, p.23):
LocalDB isn’t SQL Server Express, nor is it SQL Server Compact.
LocalDB uses the same sqlservr.exe engine as the other editions of
SQL Server, but it runs in user mode—not as a service. LocalDB is
used for offline development by tools such as SSDT to ensure that the
code you develop is 100-percent compatible with your production
SQL Server database.
If I am reading this correctly, LocalDB seems to be more like a configuration option of Express than an entirely separate product. So apparently if I download Express (or Express w/ Tools), I will have the option to install the LocalDB version, which is supposedly easier ("zero-configuration") than the full Express. (Update: with VS2012, LocalDB comes installed by default.)
One other important distinction, according to this post, is that "currently Visual Studio 2010 doesn't really work with LocalDB." (We have to use SS Management Studio instead, at least for now.)
*(The concept of "User Mode" or "User Instances" is a key distinctive of LocalDB. In fact, according to this post, "LocalDB can be seen as an upgrade of the User Instances feature of SQL Server Express." For more about User Instances, refer to the MSDN blogpost "What is a RANU?")
UPDATE - Feb 2021
LocalDB is SQL Server Express edition
From official documentation
Express edition is the entry-level, free database and is ideal for
learning and building desktop and small server data-driven
applications. It is the best choice for independent software vendors,
developers, and hobbyists building client applications. If you need
more advanced database features, SQL Server Express can be seamlessly
upgraded to other higher end versions of SQL Server. SQL Server
Express LocalDB is a lightweight version of Express that has all of
its programmability features, runs in user mode and has a fast,
zero-configuration installation and a short list of prerequisites.
The docs provides a very detailed comparison of features between editions.