The difference is in what is the usual as opposed to special.
When you say
I don't often watch TV.
The meaning is that you watch TV rarely. You might watch it once a week or so maybe. Also it is probably more common to say this as, "I don't watch TV often." So the usual state is you don't watch TV, but sometimes you do. So the exception here is that you watch TV.
I often don't watch TV.
Can be parsed as "It is quite common that I don't watch TV." In other words here the usual state is you watch TV but often you don't.
Again as an example you might not watch TV 3 days in a week and watch it the rest. Here the exception is that you don't watch TV.
As an addendum with regards to what you learned. Both of the sentences are grammatically correct, they just have a somewhat different meaning.