Learn English – Difference between ‘haven’t …yet’ and ‘didn’t…. yet’

"I have never said this" is in the present perfect tense and implies that at no time in the past have you ever made the utterance that "this" refers to. You might say this in response to an accusation that's shocking to some degree: "You told some of my friends that you hated me!", "No, you're wrong, I have never said that".

"I never said this" is in the simple past and implies that in one particular instance that is understood by the context you did not say what you're being accused of saying. "Did you tell Dale that you were going to wait for me on Tuesday?", "No, I never said that".

Note that in both constructions, it's probably more usual to use the demonstrative adjective "that" rather than "this" -- why? Because it seems to me that these phrases would most likely get used in response to a particular point by someone else said previously, and so you want to refer to that particular point.

If someone had a written phrase on paper, pushed it across the table to you, and then said "we believe that you made this statement", then you might look at it and say "No, I never said this" where "this" indicates "what's written on this paper".

• If you worked on it yesterday, stopped yesterday before finishing then, and don't want to say anything else about whether you will or can continue or not, but more likely that you are done for good (like a one-time test), then:

  I didn't finish it.

• If you worked on it yesterday, stopped yesterday before finishing, and want to imply that you are still planning on working on it, then:

  I haven't finished it.

  or with a bit more emphasis on the expectation of continuing

  I haven't finished it yet.

• The sentence

  *I didn't finish it yet.

sounds off, because "didn't" isn't particularly continuous, but "yet' implies an expectation or possibility, and so also a continuing act.