I am generating a 5e high-elf rogue for Waterdeep Dragon Heist and have chosen Message, which requires small pieces of copper wire. I've bought 2.5 foot of copper wire, cut to inches.

I have a belt pouch, but no component pouch. I have 14gp and a component pouch costs 25gp.

Is there any practical or fun differences between pouches? Are there any reasons why I would require a component pouch over a belt pouch? Storywise, would there be any problems getting into debt or get caught lifting a component pouch?

They are different. A component pouch is specialized equipment.

tl;dr The rules specifically require a component pouch which is a listed peice of equipment different from a pouch.

Rules specifically require a component pouch

The material component aspect of casting a spell specifically calls out the use of an arcane focus or component pouch.

Narrative explanation

Component pouch covers the cost of mundane components and does not require the caster to retrieve components from around their person (that might require use an object) while casting it.

Component pouch and pouch are different pieces of equipment

In the basic rules adventuring equipment, a pouch is 5sp while a component pouch is 25 gp. The component pouch takes the place of an arcane focus for a single classed spellcaster as elaborated on in the answer here.

The regular pouch does not take the place of the arcane focus, nor does it supply the mundane (no specifically cost listed) components of spells.

Can still cast message without the pouch.

Since the character directly owns the components, they can still cast the message spell. A component pouch or focus can be used in place of mundane components.

A character can use a component pouch or a spellcasting focus (found in “Equipment”) in place of the components specified for a spell.

Copper wire not consumed

The message spell does not state that the component of the spell is consumed, so a single short piece of wire should be sufficient for as many castings as you like.

One is filled with components automatically and is watertight, the other is not

The Basic Rules lists under the Adventuring Gear:

Component Pouch. A component pouch is a small, watertight leather belt pouch...

Component pouches include components

The difference between the two is that the Component Pouch includes your components and is compartmentalized and watertight. It includes all components

except for those components that have a specific cost (as indicated in a spell's description).

A regular pouch is only going to be a regular pouch and you'll have to go and get your components. The regular pouch entry also mentions the subset of itself, the component pouch:

A compartmentalized pouch for holding spell components is called a component pouch (described earlier in this section).

Once filled, they're the same thing

However, if you've gone to the trouble of finding all of your material components and put them in a standard pouch, you've effectively got a component pouch. However, should it get wet, your components may be ruined.

Character Sheets

The pregen character sheet has both pouch types. In this case, your character has both a filled component pouch and a second pouch.

The Component Pouch comes with the components.

The belt pouch is just a pouch. There is no need to buy the component pouch if you have the components already. This isn't a video game, where there is a quantitative difference between a component pouch and a pouch with components in it. It's just that when one entry lists component pouch, it comes with the components. If it is a belt pouch, it is just an empty pouch. So you could in play buy a belt pouch, put all the common components in it, then sell it as a component pouch, because that's exactly what it is. A belt pouch someone put components in and sold for 25gp.