We absolutely do notice the difference. One is voiced, the other one is voiceless. The way you articulate them is different as a result, even though there are obvious similarities as well.

I actually don't think the distinction between "thing" and "this" was pointed out when I learned English in elementary school in Germany.

They were just ambiguously referred to as the English "Tee Aitch", which doesn't help anyone.

Note they did point out the different pronunciations for each word but it was pretty much along the lines "they're the same thing but sometimes sound differently".