Preprocessing makes a HUGE difference in any type of analysis. And so you'd want to try and match these as closely as you can. Perhaps trying to use preprocessed data from CONN in AFNI or AFNI's proc.py data in CONN.

Beyond preprocessing, there are differences in how CONN and AFNI handle significance testing. CONN has options for Random Field Theory (RFT) and other methods, whereas AFNI recommends cluster-based thresholding. Similarly you'd want to match these styles.

Has it been systematically studied? There's lots of published controversy over the choices for literally every step of fMRI processing. Even packages like fMRIprep that are meant to chose a "best of best" methods have changed programs or options throughout. My recommendation is to process in whichever package and then publish your results as clearly as you can alongside the data so others can replicate (or possibly not) the results. This combined with conservative thresholds should please most journals and reviewers.

Beyond this cheerleading, not sure we can be much help other than to backup our choices in default processing options with our own publications or beliefs.