What is the difference between SQL Server 2012 Express versions?

This link goes to the best comparison chart around, directly from the Microsoft. It compares ALL aspects of all MS SQL server editions. To compare three editions you are asking about, just focus on the last three columns of every table in there.

Summary compiled from the above document:

    * = contains the feature
                                           SQLEXPR    SQLEXPRWT   SQLEXPRADV
 ----------------------------------------------------------------------------
    > SQL Server Core                         *           *           *
    > SQL Server Management Studio            -           *           *
    > Distributed Replay – Admin Tool         -           *           *
    > LocalDB                                 -           *           *
    > SQL Server Data Tools (SSDT)            -           -           *
    > Full-text and semantic search           -           -           *
    > Specification of language in query      -           -           *
    > some of Reporting services features     -           -           *

Scroll down on that page and you'll see:

Express with Tools (with LocalDB) Includes the database engine and SQL Server Management Studio Express)
This package contains everything needed to install and configure SQL Server as a database server. Choose either LocalDB or Express depending on your needs above.

That's the SQLEXPRWT_x64_ENU.exe download.... (WT = with tools)

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Express with Advanced Services (contains the database engine, Express Tools, Reporting Services, and Full Text Search)
This package contains all the components of SQL Express. This is a larger download than “with Tools,” as it also includes both Full Text Search and Reporting Services.

That's the SQLEXPRADV_x64_ENU.exe download ... (ADV = Advanced Services)

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The SQLEXPR_x64_ENU.exe file is just the database engine - no tools, no Reporting Services, no fulltext-search - just barebones engine.