This depend upon the logistic regression equation.
y(x)=1/(1+e^(-c*x))
or
y(x)=1/(1+e^(c*x))
Both equations are used to represent the logistic regression equation. Here you see value of C in both equations are same but the signs are opposite to each others.
I did not make sure, but it seems that the default reference category for dependent variable is different in both programs. Namely, if SPSS uses the level 1 as the reference, STATA seems to do the opposite and uses the level 0 as the reference. Hence one of them is modeling successes whereas the other does failures. If you fix the reference category in both programs, you will get the same results.