i believe there was an official ruling of each organization’s role to avoid conflict – Bonei Olam is more involved with medical intervention, A Time is more involved with emotional support and advice- both very necessary. I hope i am not misquoting – just trying to answer your question.
First of all I’m also in your boat about these two (with being confused about them), but as far I understand, the Rabbonim behind these org. have different halachalic opinions about some treatment used in this field. Also aTime is also giving support for people that face IF r”l which can’t be helped with treatment but with chizuk, by making Shabbatons etc. helps them to get to know other people in their situation… v’chulu…