did is past tense and when you say yet its present tense , and 'Yet' implies continuation and its used to talk about something which is expected to happen. It means ‘at any time up to now’. also It is used in questions and negatives. ,
I haven't seen it yet means -----: I have not seen this thing but there is a possibility to see it , you still be able to see that thing , its what we call " present perfect " Example : Paul has broken the vase another example : Dad hasn’t washed up yet !! but there is still a posibility to do that . it's implied that there's still an opportunity for you to "see" whatever there is to see
the first is simple past ( i did not see ) : i think you know the difference now