Reading through the Player's Handbook, I've noticed that the Sorceror, Warlock and Wizard classes can all take either an arcane focus or a component pouch. My understanding is that an arcane focus takes the place of the component pouch for most material requirements, so having the choice seems meaningless.
What are the differences between the two, and in what situations would each be better than the other?
A character can use a component pouch or a spellcasting focus (found in “Equipment”) in place of the components specified for a spell. But if a cost is indicated for a component, a character must have that specific component before he or she can cast the spell.
If a spell states that a material component is consumed by the spell, the caster must provide this component for each casting of the spell.
A spellcaster must have a hand free to access a spell’s material components—or to hold a spellcasting focus—but it can be the same hand that he or she uses to perform somatic components.
System Reference Document, p. 102
Either the component pouch or the arcane focus can be used to supply material components for the sorcerer, warlock, and wizard spells, provided the material components are neither costly nor consumed.
Additionally, neither of them interfere with somatic components, allowing a spellcaster to keep his other hand occupied with a weapon, shield, grappled enemy, torch, etc.
a spellcasting focus, such as a holy symbol, can be used only for the spells from the class associated with that focus
System Reference Document, p. 58
If you wish to cast spells from different classes, depending on which ones, you may prefer to have a spell component pouch, as it will function with every class.
Additionally, Eldritch Knights, Arcane Tricksters, and rangers do not have a class ability the allows the use of any spellcasting focus, so they must use a component pouch or track components individually.
The component pouch (25 gp) is more costly than an arcane focus (5-20 gp), and crystals and wands (1 lb.) are lighter than component pouches (2 lbs.). (SRD, p. 69)
An arcane staff may be usable as a quarterstaff in combat, and an arcane staff might possibly also be a druidic staff.
A component pouch could also serve as a pouch to hold costly or consumed spell components, but those must be tracked individually as the pouch cannot be assumed to have them at all times, as it can for non-costly, non-consumed components.
Standard belt pouches are very cheap (5 sp, 1 lb.), and should be capable of storing costly and consumed components unless you have a wide variety of them, in which case a spell component pouch may be more appropriate.
It should take the same number of "interact with an object" uses to:
as it does to:
Neither drawing a wand nor pulling a spell component out of a component pouch are listed as example interactions in the "Interacting with Objects Around You" block (SRD, p. 92), but similar actions are listed: "draw or sheathe a sword" and "fish a few coins from your belt pouch".
It's up to your GM and your table to decide if such activity requires zero, one, or two object interactions, but I don't see any reason why the number should differ between component pouches and spellcasting focuses.
As you have noted, the primary function of both an arcane focus and a component pouch is to replace the non-costly material components of a spell.
The biggest difference was clarified in the rules of spellcasting sage advice. The following question was asked:
If a spell's material components are consumed, can a spellcasting focus still be used in place of the consumed component?
Nope. A spellcasting focus can be used in place of a material component only if that component has no cost noted in the spell’s description and if that component isn’t consumed.
This is important for spells such as Protection From Evil and Good which have a material component without a specific cost, which is consumed. This spell is impossible to cast with only a focus because the material component is consumed; as clarified in the sage advice. However, you can cast this spell with a component pouch, since you can retrieve these non-costly items from within it.
Whenever a spell calls for a consumed material component without a specific price attached, the focus will fail you, but the pouch will not.
There very little functional difference between the two for a single-classed spellcaster, as long as you're not dealing with expensive or consumed material components. Expensive components are required as described, and consumed components may be too, if you adopt certain Sage Advice suggestions at your table.
Either one serves to meet the "Material Components" requirement for a spell. For spells with material components, the caster must have a free hand to access Material components (PHB p209), though it can be the same hand used for Somatic components. So... in order to make use of a focus, the character must have it in hand.
However, they have an effect on action economy. You only get one free object interaction per turn, either putting something away or taking it out. Getting your focus into your hand uses that up. A component pouch, on the other hand, is something you wear (it goes on your belt, PHB p 151).
For pure caster characters, the action economy isn't that big a deal, but for a multi-classed spellcaster, it is if they don't all use an "arcane focus". You only get one free object interaction a turn. If you want to switch focuses, that means you have to waste an action putting one focus away and drawing the other. Your only alternative is to drop the first focus (dropping something is a non-action). However, both classes can share the same component pouch, and as long as you have a free hand, you can use it.
Worth special note is a holy symbol, which specifically states "worn" is sufficient (PHB p 151). For the classes that can use them (cleric and paladin), and spells that also don't have a somatic component, you don't need a free hand. The specific case of the holy symbol trumps the general case of needing a free hand for material components. Of course, if doesn't help you with somatic component spells without material components (you only get to double up the focus-hand with the somatic-hand if you're actually using it as a focus), so a cleric or paladin may need to put his weapon away using the single free object interaction. The War Caster feat lets you get around the issue for somatic components, by the way.
As you've said, for any material component without a cost, a Focus can substitute. The only mechanical difference is that a component pouch costs twice as much as the cheaper foci. However, flavor-wise, there's quite the difference. A caster using their focus has quite a different air about them than one pulling the components from their bag. And through creativity, you can turn that into a mechanical difference.
Say that the party is arrested and thrown into a prison, but you had a bit of time where you knew it was likely. Your jailors will certainly confiscate your focus and/or your material pouch, but they'll probably think less of the little bit of wood and string you took out of your pouch for Unseen Servant. Or perhaps your DM is a cruel one, and decides the component pouch runs out of components after a while, and you suddenly find yourself struggling to cast anything. As cyphra just stated, this is dipping into "why materials to begin with", though.
Even if you do, but within the sorcerer, warlock, wizard list, still no difference.
If you multiclass with anything else, you can use the component pouch for the spells of both.
From focuses you would need two, one for each class.
Other than the obvious (different prices, different appearance), there is not functionally a lot of difference outside of fluff, especially if you look at the discussion regarding your exact question in GitP.
What might also be relevant is why materials are required in the first place, when keeping them/getting them is trivialized by these two items.
