From a little research I did earlier for a question about 分かる vs 理解する, it seems like 知りません means “I don’t know” specifically, while 分かりません can also mean “I don’t know,” but has various other meanings/usages as well.
Umm…though I don’t have much IRL Japanese experience…just lots and lots of anime mostly. From which I can also say, I can definitely see how 知りません seems colder somehow…like it’s more final/definite–“I don’t know,” whereas maybe 分かりません is closer to, “I’m not aware of it.”