Are you sure variables are added in the same order?  This usually has a very strong effect on the outcome of approximation.

Thank you Alexei,

I have just run it once again. All the results from the variables are the same except from the intercept....

                     Estimate     Std. Error    t value      Pr(>|t|)

(Intercept)    20.162          1.283         15.711     < 2e-16   (R 3.1.2)

                    19.493          1.534         12.707     0.000    (SPSS 20)

I have a suspicion that  rounding plays a role because the two results are very close...

Yes, the results are very close, so you probably should not worry too much about it.  The least squares method is never very precise, especially when your data is not normally distributed -- and then even things like the sequence of processing data points starts to make a difference.  Did you check data normality and homoscedasticity?  If you can transform the data to improve these two parameters, the precision of your estimate will shoot up dramatically.