I think the difference is only the context. As in your picture "that many" is used to express surprise at the quantity. "So many" might be used to express exhasbiration that the number is so big.

'That many' is referring to a particular quantity mentioned during the conversation. A slightly different way to the above example could be: Alex: "I've been to 52 countries" Kate: "Wow that's loads, I wish I had visited that many" "So many" is a bit more versatile. Kate could have also responded as follows: "Wow you've been to so many countries!" She could also start a new conversation mentioning a new quantity: Kate: "I went to John's place yesterday and, wow, he has so many cars!" But she couldn't say "he has that many cars!" because 'that' has to refer to something that has already been mentioned in the sentence or earlier in the conversation. The usage of the 2 phrases is pretty nuanced and more complicated than what I've outlined, but I hope I've provided a useful intro explanation :)

In your example, there isn’t really any difference. In a statement however, “so many” is often synonymous with “quite a few”. Note that “quite a few” means “lots and lots”, whereas “quite few” means the opposite (i.e. not many at all). Keep in mind that I didn’t invent this crazy language — I just teach it :)