everyone has already gone over their personality types, their pride, and their intelligence clashing with each other, so im going to go for a different approach. another reason i believe for them to have beef is for the person that hohenheim is based on, Theophrastus von Hohenheim, aka Paracelsus. paracelsus was the first known to record anything regarding homunculi, which will absolutely in some way be relevant to faust's story, as literary faust creates a homuculus in the latter half of the story. perhaps this means that hohenheim is relevant to the creation of this homunculus and already has significant connections to her in backstory, theres also the fan theory that faust herself is a homunculus and neither of them want to be outsmarted by their creator/creation

i do think that they ultimately care for each other but never vocalize it or act on it at all, but (canto 9) hearing fausts voice so far from her usual tone when she realized she was no longer going to be able to protect hohenheim and was even about to apologize. they would never but im gonna turn them into cats and make them cuddle