"Tamtej" is feminine (e.g. "tamtej kobiety" - that woman), while "tamtego" is either masculine ("tamtego mężczyzny") or neutral ("tamtego dziecka" - that child).
If you want to say "You do not know that man.", and for example you speak with someone about your mutual acquaintance, would it be more natural to say "tego" instead of "tamtego"? As I have noticed by reading Polish, they tend to use "this" much more often and in larger context than the English speakers do. Similar things happens in Croatian, my native language, also.
It would be great if a native speaker could tell something more.
I'm from Poland! :) and I think you may be right. We say "tego" more often in everyday situations, but it's pretty hard to say if it's more natural to say "tego" instead of "tamtego" in your example. The difference between these two words is very small and even for me - a native speaker - it doesn't really matter if you say "tego" or "tamtego". Personally I would say that both of them are correct if I had to translate your sentence. "You don't know that man" = "Nie znasz tego/tamtego mężczyzny". But it all depends on how you want to translate it. If literally, then I would recommend to say "tamtego". I hope it helped you :) If you have some more questions, you can ask me ;)