I would like to know what each one actually does. If you have the animal friendship spell, do you need any animal handling?
We can look at the SRD for answers.
When there is any question whether you can calm down a domesticated animal, keep a mount from getting spooked, or intuit an animal’s intentions, the GM might call for a Wisdom (Animal Handling) check. You also make a Wisdom (Animal Handling) check to control your mount when you attempt a risky maneuver.
This spell lets you convince a beast that you mean it no harm. Choose a beast that you can see within range. It must see and hear you. If the beast's Intelligence is 4 or higher, the spell fails. Otherwise, the beast must succeed on a Wisdom saving throw or be charmed by you for the spell's duration. If you or one of your companions harms the target, the spells ends. At Higher Levels. When you cast this spell using a spell slot of 2nd level or higher, you can affect one additional beast for each level above 1st
Animal Handling allows you to influence an animal's behavior when the animal is either domesticated, or is owned by you. If it isn't domesticated or owned by you, according to the book you can only try to intuit its intentions. RAW, you can't use animal handling to calm down an attacking bear in the wild, but a successful animal handling check might tell you whether or not it wants to eat you or is just defending its den.
Animal Friendship can be used on any animal regardless of its background or pedigree. You can absolutely use this spell on the wild bear you meet out in the woods, and if it's successful then you can convince the bear that you mean it no harm (even if it previously thought you did). This is different than Animal Handling because if the bear thought you meant it harm, a successful Animal Handling check would not convince it otherwise.
As to whether or not you need Animal Handling if you already have Animal Friendship, keep in mind that spells cost resources and can only be used a certain number of times a day. You don't want to have to cast a spell every time you ask your horse to charge into battle or every time you want to put barding on it. Also, RAW Animal Friendship doesn't tell you anything about the animal's intentions, so if you ever find yourself in a position where you want to know an animal's intentions, you'll want Animal Handling in addition to Animal Friendship.
animal friendship is a charm spell.
Animal Handling is the ability to calm a domesticated animal, keep a mount from getting spooked, or intuitively predict and animal's intentions. You make a wisdom check when attempting an action with an animal.
What is the limit of actions a charmed being will do? The spell is broken if you or your party harm the charmed being. IN the games I've played as a PC and as DM, that harm included forcing the charmed being to harm itself. (This means, make the charmed being jump off a cliff, run into a wall, etc., but does not include the charmed being fighting with the party against a common enemy.)
Animal handling would be controlling the animal, and checks are made whether successful. For example, getting a horse to jump a dangerous jump despite its limited wisdom telling it to not jump.
Now, here is an example where you would need animal handling on an animal that has animal friendship on it: You have a bear charmed. You mount it to ride it. You go up to full speed. Now, the bear, charmed will obey and run at full speed. However, it is not trained to run in a manner to keep you on its back, so you will probably fall. Now, having animal handling would give you the knowledge to know how to guide the bear to run in a manner that you are less likely to fall off.
Animal Friendship spell specifically names a type as the target (beast) with INT less than 4, to be charmed by you and become friendly to you and your allies. Animal Handling skill applies to a broad array of creatures, but with limited effects. For example, you cannot target a creature type: Monstrosity with Animal Friendship, but you could still make an Animal Handling check to attain whichever result is acceptable for that situation. (You can't charm an Owlbear with that spell, but you can use the skill to gain information on how to entreat it)