Coligno seems to have this pretty much right. I remember initially only being taught tāku and tōku and only later on once those were cemented were we taught we could use taku (which is neutral). I always felt it was probably only used in formal situations. My Dad knows this kind of thing well, so I might ask him and check back in here with his answer.

The difference is to do with dominant and subordinate possession (like the difference between "a" and "o" meaning "of") "tāku" means "my" when I am active or dominant to the thing possessed "tōku" means "my" when I am passive or subordinate to the thing possessed e.g. tāku tamaiti "my child" cf. te tamaiti a te wahine "the woman's child" tōku waka "my canoe" cf. te waka o te wahine "the woman's canoe" In "taku" there is no distinction between dominant and subordinate possession, but the use of this word is limited to certain parts of the phrase. I still have to understand this one fully myself.