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.
Dear @MaDGab Is this example I rarely don't watch TV for a week. correct? Rarely and don't in one sentence? I think it isn't correct. Why is negative don't before 'rarely'? I just ask. – Ľubomír Masarovič – 2017-02-28T08:36:52.870
@ĽubomírMasarovič "I rarely don't watch TV for a week." is absolutely fine. You can't switch the order around here though "I don't rarely watch TV for a week." is possibly dialectal, but ungrammatical for a standard speaker IMO. – DRF – 2017-02-28T08:49:26.317
@ĽubomírMasarovič Yes, it's correct. (not sure exactly how helpful it is in the context of the OP's original Q though - even though I upvoted) – Araucaria - Not here any more. – 2017-02-28T08:50:21.057
@Araucaria any clue if "I don't rarely watch..." is ok in some dialect? I feel I've heard this sort of inversion but can't place it. – DRF – 2017-02-28T08:58:41.563
1@ĽubomírMasarovič It's a bit of a contrived example, only the sort of thing you'd hear in informal speech when someone hasn't thought their sentence through properly. It's technically correct but there are much less clumsy ways of saying it, like listed below. Think of it as "I rarely (don't watch TV for a week)", that is "not watching TV for a week" is the thing you do rarely. – Muzer – 2017-02-28T09:47:27.937
@Muzer I have read this article link. There is one important sentence - We use them without not. That is the reason why I am a bit confused when I see in one sentence not and rarely together.
– Ľubomír Masarovič – 2017-02-28T11:12:53.810@ĽubomírMasarovič In some languages you would always use such sentences with "not" and they would mean the same as "he rarely watches TV for a week". In English you don't generally use them with "not", but you CAN technically still use "not" to negate the meaning, and you'll probably be understood - but it sounds clumsy and as I said, isn't the sort of thing you would plan to say. So as another example, "I rarely eat dinner" would mean that you don't eat dinner often, but "I rarely don't eat dinner" while sounding clumsy is a valid way of saying "I usually eat dinner, but not always". – Muzer – 2017-02-28T11:20:16.643