Rule of thumb: You can always omit that after the reason (i.e. the reason (that)). Put it another way, wherever you can say the reason that, you can omit that.
That vs. Why: You can use that (which is optional) in place of why but only in defining clauses, and make sure that you use reason not reasons (plural) if you want to use a that-clause. (See Cambridge Dictionary Online's note below.)
Question: [W]hat are the similarities or differences between the two sentences below?
- He didn't tell me the reason that he wore a polka-dot dress.
- He didn't tell me the reason he wore a polka-dot dress.
Answer: There is no difference in meaning whatsoever.
Macmillan lists these uses of reason, among others:
1 [countable] a fact, situation, or intention that explains why something happened, why someone did something, or why something is true
reason for: The police asked her the reason for her visit.
reason for doing something: Could you explain your reasons for choosing this particular course?
reason why: The reason why so many people caught the disease is still not clear.
reason (that): The reason these cars are so expensive is that they are largely built by hand.
Cambridge Dictionary Online lists reason why, reason that, and reason + to-infinitive.
Reason why We use reason why before a clause. We often omit why, especially in statements:
The reason (why) I didn’t contact you was that I was only in town for a few hours.
Reason that We use reason that before a clause. We often omit that, especially in statements. Reason that is less common and slightly more formal than reason why:
The reason (that) we need new guidelines is that the present ones are just not working.
We do not use reason in the plural with a that-clause:
There are several reasons why I don’t like the book.
There are several reasons that...
Reason + to-infinitive We can use reason with a to-infinitive:
There’s no reason to be suspicious – everything, is perfectly normal.
Longman English Grammar (by L. G. Alexander) mentions reason why, reason for which, and reason that shortly.
That's the reason (for which) he dislikes me.
That's (the reason) why he dislikes me.
My success in business, (the reason) for which he dislikes me, has been due to hard work.
My success in business, the reason why he dislikes me, has been due to hard work. (The reason cannot be omitted before why.)
1.38.4 ('That') in place of 'when', 'where', 'why'
That is possible (but optional) in place of when, where and why but only in defining clauses:
That wasn't the reason (that) he lied to you ((That) can be replaced by why or for which)
†Even though you've asked specifically for the exclusion of reason why, I decide to keep all alternatives--reason why, reason that, reason for, reason for doing something, reason to do something--in the above quoted texts, because it could, I believe, make this question more useful for the future reader.