The two-sample Kolmogorov–Smirnov test is a common approach for comparing two distributions. The original version applies to univariate distributions, but multivariate alternatives exist. Wikipedia:
A distribution free multivariate Kolmogorov–Smirnov goodness of fit test has been proposed by Justel, Peña and Zamar (1997). The test uses a statistic which is built using Rosenblatt's transformation and an algorithm is developed to compute it in the bivariate case.
The Kolmogorov–Smirnov test statistic needs to be modified if a similar test is to be applied to multivariate data. This is not straightforward because the maximum difference between two joint cumulative distribution functions is not generally the same as the maximum difference of any of the complementary distribution functions. Thus the maximum difference will differ depending on which of $\Pr(x < X \wedge y < Y)$ or $\Pr(X < x \wedge Y > y)$ or any of the other two possible arrangements is used. One might require that the result of the test used should not depend on which choice is made.
One approach to generalizing the Kolmogorov–Smirnov statistic to higher dimensions which meets the above concern is to compare the cdfs of the two samples with all possible orderings, and take the largest of the set of resulting K–S statistics. In d dimensions, there are 2$^d$−1 such orderings [With your 2 dimensions, this is 3 orderings]...Critical values for the test statistic can be obtained by simulations, but depend on the dependence structure in the joint distribution.
This test would be applied to the data that you used to produce the scatterplot. I'm not sure if it can be performed in SPSS without scripting it yourself, which is feasible. I think @Epaminondas' comment covers the difference between distribution tests and mean differences tests such as MANOVA.
9. Justel, A., Peña, D., & Zamar, R. (1997). A multivariate Kolmogorov–Smirnov test of goodness of fit. Statistics & Probability Letters, 35(3), 251–259. DOI:10.1016/S0167-7152(97)00020-5.