After some digging, I found this stackoverflow post which really helped explain the difference.
cast_assoc/3 is only used in the changesets of schemas that have has_one or has_many. It invokes the cast/3 validation in the changeset of the associated schema.
assoc_constraint/3 is used in the changesets of schemas that have belongs_to. It checks to make sure the parent schema is in the database so an orphaned record isn’t inserted.
foreign_key_constraint/3 is used similarly to assoc_constraint/3, except that it uses the database to validate the foreign key constraint, whereas assoc_constraint/3 uses the Ecto schema definition.
The reason I was having issues inserting records with associations was because of the way I defined the association. Ecto was trying to insert the parent record along with the child record. As noted in that linked article, the correct way to insert a child record with a parent association is to either build the the association using build_assoc/3 from the parent record, or, in more complex casts, put the parent foreign key in the parameters of the child record and pass that to the changeset.
See one, do one, teach one 
