Logically they mean the same thing - if there isn't one thing, there aren't many things. But they can both be used in different contexts. There's no other choice is usually used in 'yes/no' situations. We'll have to buy a new one - there's no other choice. There are no other choices can be used to show that the limit of choices has been reached. We can use company A, B or C but there are no other choices. You could also replace choice(s) with option(s).

Generally when we say "no other choice" it means that we have tried various previous options, therefore when we say "no other choice" we are using the word choice in a singular form, expressing that we do not have a single choice. Choices are the plural form of the word, which does not go hand in hand with the words "no other".