Z-test is for large sample test for mean
Chi-square test is for test for single variance
The Z-test is for testing a mean WHEN we know the pop variance, or we approximate it via the sample variance,
THEN it's a large sample situation where you're using the CLT.
Or a t if n is small and we have normality and we are using s instead of sigma.
Chi-square test is for testing a single population variance
But there are so many tests.
You use the chi-square in goodness of fit tests too...
You really need a more precise question here.