The in game functional difference between the storage and companion cubes is, none. Functionally both can push buttons, be stacked, block lasers/turrets/pellets.
Both dissolve in empancipation grills and water and pits.
The only difference is one is blue, one is pink with a heart.

In Portal 1, the Companion cube is used quite extensivly in its test chamber.
Used to climb stairs, block pellets, redirect pellets, push buttons, and ultimately is incinerated.


In Portal 2 is it used to block lasers and hold buttons, a diminised role to be sure compared to the first game.


While any of the above could also be done with a storage cube, the difference is the companion cube must be used from begining to end. Its never left behind, never replaced. (Excluding Glados's torture fizzles in portal 2)

Due to this reusing, a player is trained by the campain to see the companion cube as something special. Something they'll need over and over.

As a map designer this is something you can take advantage of. While a lot of people like the companion cube, spamming it overall diminishes its impact.

In this sort of map, the player is given plenty of cubes, and it doesn't matter if a cube is lost.

In this sort of map the player is given only one cube. And loosing it has significantly bad consequences.


To use the compaion cube in a campain manner and use a players expectation

• Make it significant and needed for the majority of the map
  
• Make it the only cube (Or multiple companion cubes)

Making loosing a companion cube having significant consequences is a good idea as it renforces to the player its importance and that they have to keep it around.


If a map makers truely wants to emulate the companion cube in the campain, keep in mind in both games neither companion cube runs the risk of being last. Either not getting near a fizzler, no working fizzlers, and no pits or other ways to loose it.
It no work!

The other significance is an easter egg, the player can't use it but its there to find.