SQL Server 2014 Replication - The process could not connect to Subscriber 'xxx'
Community Wiki answer generated from question comments by Brandon Williams 

In the tutorial, the repl_distribution account will need to be added to the subscription database as db_owner.

In the New Subscription Wizard, on the Distribution Agent Security page, you specify a Process Account, which would be the repl_distribution account that was created on the Publisher. Now, further down on the page is a section 'Connect to the Subscriber'.  Choose 'By impersonating the process account'.

If the account exists at the Subscriber with the same username and password, and appropriate permissions, it will work.  This is known as pass-through authentication.  Alternatively, you can use SQL authentication to connect to the Subscriber.

Response from dthree:

I tripled checked all of the security settings to make sure it was all set up right, and in that I found that I shared and set the security settings on the wrong folder for the repldata snapshot. Fixed that, and between both fixes it worked.