They're functionally equivalent for a single-classed spellcaster.
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.
The component pouch is more useful for a multiclassed spellcaster.
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:
- pull some guano out of a pouch
- use it to cast fireball
- put it back in the pouch
as it does to:
- pull out a wand
- use it to cast fireball
- put it back in your backpack or wand-holster.
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.