Here are t-scores and p-values from a set of t-tests that I recently conducted in SPSS and in Stata:
Group 1 unweighted
t = 1.082 in SPSS (p = 0.280)
t = 1.082 in Stata (p = 0.280)
Group 2 unweighted
t = 1.266 in SPSS (p = 0.206)
t = 1.266 in Stata (p = 0.206)
Group 1 weighted
t = 1.79 in SPSS (p = 0.075)
t = 1.45 in Stata (p = 0.146)
Group 2 weighted
t = 2.15 in SPSS (p = 0.032)
t = 1.71 in Stata (p = 0.088)
There was no difference between unweighted SPSS p-values and unweighted Stata p-values, but weighted SPSS p-values fell under conventional levels of statistical significance that probability weighted Stata p-values did not (0.10 and 0.05, respectively).
John Hendrickx noted some problems with weights in SPSS:
One of the things you can do with Stata that you can't do with SPSS is estimate models for complex surveys. Most SPSS procedures will allow weights, but although these will produce correct estimates, the standard errors will be too small (aweights or iweights versus pweights). SPSS cannot take clustering into account at all.
Re-analysis of Group 1 weighted and Group 2 weighted indicated that t-scores in Stata were the same as t-scores in SPSS when using the analytic weight option [aw=weight] and the importance weight option [iw=weight].
SPSS has another issue with weights, indicated on the IBM help site:
If the weighted number of cases exceeds the sample size, tests of significance are inflated; if it is smaller, they are deflated.
This means that, for significance testing, SPSS treats the sample size as the sum of the weights and not as the number of observations: if there are 1,000 observations and the mean weight is 2, SPSS will conduct significance tests as if there were 2,000 observations. Stata with the probability weight option treats the sample size as the number of observations no matter the sum of the weights.
I multiplied the weight variable by 10 in the dataset that I have been working in. For this inflated weight variable, Stata t-scores did not change for the analytic weight option, but Stata t-scores did inflate for the importance weight option.
UPDATE (April 21, 2014)
Jon Peck noted in the comments that SPSS has a Complex Samples procedure. SPSS p-values from the Complex Samples procedure matched Stata p-values using probability weights:
The Complex Samples procedure appears to require a plan file. I tried several permutations for the plan, and the procedure worked correctly with this setup:
UPDATE (May 30, 2015)