A: Is he going to go into coma?
B: I shouldn't think so. /
I don't think so.

What would be the difference between these two responses?

Thank you

To my ear, I shouldn't think so (or I wouldn't think so) is just a fancier way of saying I don't think so. But unlike the simpler I don't think so, it suggests that I find no reason to think that the proposed situation will occur or that I would be surprised if the proposed event happened. Perhaps there is even a shade of not wanting the proposed event to happen.
By the way, this is not the should of advice but of expectation. It's I don't expect so. Not It's not advisable for me to think so.

Back in the Middle Ages while in high school, I was taught that should is used with the first person, would with the 2nd and 3rd, and using the opposites for emphasis. The same with will and shall.

So, when General MacArthur said, "I shall return (using shall with the 1st person, trying to be grammatically corect), he should have said, "I will return", for emphasis.

As I said, this distinction has largely disappeared in AmEng.

I think this now is a case of convention rather than rule, as things go today.
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No difference in import. B1 sounds more BrE to me, and slightly more confident, but I'm no expert.
 CalifJim's reply was promoted to an answer.
I wonder if I want to use "should" in the same scenario but in the context of "should of advice", 
do I say "I shouldn't have thought so" as in "I have already thought that he is going to go into coma, but it's not advisable for me to think so"
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 Philip's reply was promoted to an answer.