Hello Juliana. I do not use the PROCESS macro. But in order for anyone to help you, I think you will have to post your command syntax for PROCESS and for the REGRESSION command in SPSS. If you could upload a sample dataset that illustrates the problem, even better. Then someone who does use PROCESS will be able to duplicate the problem, and perhaps figure out what is causing the discrepancy. HTH.
I just tested this with my dataset and the results were very similar. I'd recommend taking a look at Hayes' Introduction to Mediation, Moderation, and Conditional Process Analysis 2e (2018) from the Guilford Press. Hayes does address output from different statistical packages.