I ran the Assembly Binding Log Viewer against on both the successful and failing system. It doesn't record any binding exceptions on either system, and the logs look pretty similar. Here's the log for binding the Core DLL:
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*** Assembly Binder Log Entry (10/30/2016 @ 1:46:13 PM) ***
The operation was successful.
Bind result: hr = 0x0. The operation completed successfully.
Assembly manager loaded from: C:\Windows\Microsoft.NET\Framework\v4.0.30319\clr.dll
Running under executable C:\Users\Administrator\Desktop\myapp\myapp.exe
--- A detailed error log follows.
=== Pre-bind state information ===
LOG: DisplayName = PcapDotNet.Core, Version=1.0.4.25067, Culture=neutral, PublicKeyToken=06a20bc2fabb1931
(Fully-specified)
LOG: Appbase = file:///C:/Users/Administrator/Desktop/myapp/
LOG: Initial PrivatePath = NULL
LOG: Dynamic Base = NULL
LOG: Cache Base = NULL
LOG: AppName = myapp.exe
Calling assembly : myapp, Version=1.0.0.0, Culture=neutral, PublicKeyToken=null.
===
LOG: This bind starts in default load context.
LOG: Using application configuration file: C:\Users\Administrator\Desktop\myapp\myapp.exe.Config
LOG: Using host configuration file:
LOG: Using machine configuration file from C:\Windows\Microsoft.NET\Framework\v4.0.30319\config\machine.config.
LOG: Post-policy reference: PcapDotNet.Core, Version=1.0.4.25067, Culture=neutral, PublicKeyToken=06a20bc2fabb1931
LOG: GAC Lookup was unsuccessful.
LOG: Attempting download of new URL file:///C:/Users/Administrator/Desktop/myapp/PcapDotNet.Core.DLL.
LOG: Assembly download was successful. Attempting setup of file: C:\Users\Administrator\Desktop\myapp\PcapDotNet.Core.dll
LOG: Entering run-from-source setup phase.
LOG: Assembly Name is: PcapDotNet.Core, Version=1.0.4.25067, Culture=neutral, PublicKeyToken=06a20bc2fabb1931
LOG: Binding succeeds. Returns assembly from C:\Users\Administrator\Desktop\IGMPSpeedTest\PcapDotNet.Core.dll.
LOG: Assembly is loaded in default load context.
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Compare this procmon output from a functioning system (http://i.imgur.com/otBjEna.jpg) with the same relevant section on a system that exhibits your symptom (http://i.imgur.com/5IBEMqj.jpg). You'll notice the msvcr120.dll file is missing; this is the 2013 runtime.
Here's the download I used to solve my problem: https://www.microsoft.com/en-us/download/confirmation.aspx?id=40784.