Con images are weighted combinations of beta images created by the contrast you specified. T images are t-statistic maps, which means they’re basically the con maps divided by the variance maps (although there’s probably a square root or degrees of freedom in there that I don’t remember offhand).
It’s been a long time since I used SPM, but I believe that it only does one-sided statistical tests, so that would explain why the T map only has positive and zero values. In order to get the appropriate stats for the voxels with negative values in the contrast map, you may need to recompute the inverse contrast (e.g., B > A in addition to A > B), or there may be an option to run a two-sided test.
If you’re using nilearn’s get_clusters_table, then you should use the spmT map, because the beta values in the con maps are not really useful on their own.
Someone who is more familiar with SPM will probably be able to answer your questions in more detail, but I hope my response is helpful regardless.