SPSS has, I believe, three parameterizations via GenLin (canonical, log, and identity). Also, the last time I checked it differed in Fisher's iteration application in that it simply would stop after 5 or 6 (can't recall), but this isn't usually a problem. A bigger issue is that Stata and SPSS differ with respect to R in that both can converge outside of the parameter space. Part of this is due to initial values, at least with the GLM in general (most software packages use some IRLS, although some use null vector or others, but I am fairly certain that both Stata and SPSS use an IRLS default). Despite the sophistication of current IRLS methods compared to, well, Euler's method and those learned in calculus courses, iteration of any function that uses derivatives or partial derivatives (in particular, here the Hessian matrix) will be extremely sensitive to small changes in the application of each iteration. For Stata (and R, SAS, and others) the IRLS is the MLE method (or at least it produces the alpha estimates), and Stata has at least to NB-specific parameterizations (see attached scanned and cropped image from Hilbe's Negative Binomial Regression (2nd Ed.)

I could go on describing differences in various ways and at various levels, but it would be easier and more productive if you could post/attach/upload some file (even a screen capture) with the actual output.

 Thanks again for your effort Andrew. In the file you can see the output and the differences. Would it be problematic to take the model computed with the glm...family(nbinomial 1) ?

I can't think of a reason why it wouldn't given the information (and the fact that in general any statistical/computational software package that is at least fairly popular and comprehensive- R, SAS, MATLAB, etc.- is better than SPSS, which I maintain is just a glorified version of Excel). It wasn't until v. 15 that SPSS included GLM procedures, while Stata offered included negative binomial regression in its GLM package in the early 90s, over a decade earlier.