Curious if there's any in-game difference between the factions other than flavor text.

Is there any incentive to courting specific factions over others? Is there any incentive to playing neutral? Should I try and specifically curry favor with any in particular, or just go with the flow and see how things turn out?

Some factions give nice items if they like you. That costs favor points. Some factions are better than others. And some demands are easier to meet than others. You can also get favor points when you sell stolen faction items on the black market sometimes.

Armor/weapon developers is a good choice as some things are more situational than others but you always need good equipment. Say, FoxKin, if you are primarily anti-UFP.

Keep in mind, that even if you have good relationship some factions will not work with you if some other faction wants you dead or even favors you.