Those terms are descriptive, and do not explain where the mechanical difference between them lies. Inclusion of either phrase is not "the reason" the effects they describe exist.
When looked at overall, I think the distinction becomes pretty clear. In the entire Appendix A of the 5e PHB, there are exactly four occurrences of the phrase can't move:
- A description of the Paralyzed condition
- A description of the Petrified condition
- A description of the Stunned condition
- A description of the Unconscious condition
In each of these conditions there is additional description indicating that, along with any of those conditions, a creature is also Incapacitated:
An incapacitated creature can't take actions or reactions.
So it's true that the character not only cannot traverse any squares' worth of distance (Movement, with a capital "M") but also can't shift their position in any way at all (that's the reason for automatically failing Strength and Dexterity saving throws, and why attack rolls against them always have Advantage). Characters with any of these conditions cannot take any actions or reactions, at all, because they are also incapacitated.
The incapacitated condition is only referenced seven times in Appendix A: twice in describing incapacitated itself (the section heading and its description), once when describing one way that the grappled condition may be ended, and then once each in the descriptions of the four conditions in which the phrase can't move is used.
In the same appendix, there are two instances of the phrase speed becomes 0:
- A description of the Grappled condition
- A description of the Restrained condition
These conditions indicate a situation in which a character cannot traverse any squares' worth of movement but can still move enough to potentially take Actions (like attempting to break a grapple, or wiggle out of ropes tied to restrain the character). Characters with these conditions can take actions, but cannot move more than 0 feet.
If you are chained to a dungeon wall, or someone strong is holding you down, it is those specific obstacles which stop you from moving into another square. It doesn't matter if you are wearing (or put on) Boots of Striding and Springing, because the extra movement speed it offers doesn't deal with the obstacles at all. That's why your speed becomes 0 and cannot be increased.
If you are paralyzed, petrified, stunned, or unconscious, you are necessarily also incapacitated, which precludes any Action of any type (indeed, the book is even more expansive in using the lowercase a for action).