The tests are for different questions.
Depending on the type of chi-square test, you are probably only testing if there are significant proportional differences across the eight (2x4) categories simultaneously. So you might get significance, for example, if any 2 combinations of categories is has a smaller N than each of the the other six combinations of categories.
Proc glimmix with the multinomial option will test if your binary predictor is significantly better at predicting each of three of your outcome categories relative to a fourth outcome (reference) category. So here the question is if your binary predictor is associated with a significantly higher or lower probability of being being in one, two, or three of your outcome categories compared to its probability of being in the fourth reference outcome category.
