Can you give the grammatical rule that distinguishes between these two cases?

I found this on the Web and thought it was worth sharing. English is a difficult language to learn for numerous reasons. For example, there are times when it is permissible to have a subordinate clause without a conjunction.

Consider the following sentence:
The dog that Mary saw was a terrier.
It is permissible to drop “that” to get:
The dog Mary saw was a terrier

Now consider this sentence:
The dog that had a white collar was a terrier.
It is incorrect to drop “that” to get:
The dog had a white collar was a terrier.

Do you see what distinguishes the two cases?