I have a hard time figuring out the difference or knowing when to use what.
I haven't seen a single animal related thing that doesn't require animal handling (or at least can be substituted with it), i.e. caring, healing, breeding, training, hawking,...
Apparently for all the activities that do not directly involve handling animals you do need specific abilities like bargain (buy and/or sell tamed/trained animals), leadership (run mews including lead assistants), etc.
Can anyone explain what the difference is, why have profession: Falconry if you already have animal handling?
I figure Profession: Falconer can substitute for Animal Handling with just a few animals, plus you can use it to show nobles how to handle the falcons better than a trainer of the same skill level, know the rules of the games (like how long a chain or rope is typical in wolf baiting if you're in the east), etc.
Looking at the specifics of falconry, I would allow Animal Handling for "raising, tending, grooming, and healing" raptors in a way, which does not interfer with making them into falconry birds. But already this training IMO requires Profession: Falconry, as would specialized knowledge of raptors, their physiology, needs of food, habitat, life, hunting habits, and such. Also handling the falconry birds properly at a noble hunt or at court is more than Animal Handling: the duke expects his falcon to be ready and eager to hunt, not to soil his glove.
In detail, such overlapping skills need some discussion with the troupe before they become important.