While reading through the conditions described in Appendix A of the 5e PHB, I realised, that some of them say a creature "can't move" (e.g. Stunned, Unconscious), while others state that its "speed becomes 0" (e.g. Grappled, Restrained).
Can anyone explain to me the mechanical distinction between these two phrases?
Under some conditions, a speed that has "become zero" can be increased magically, as discussed in this question.
If you "can't move", then changes to your base speed have no effect.
Also, "can't move" does not imply "can't take an action". This is indicated by the fact that all conditions that use the phrase "can't move" include the condition Incapacitated in their description, which specifically prohibits actions and reactions.
The spell Magic Jar (PHB p.257) gives an example of a case where you can't move, but you can take an action:
You can't move or use reactions. The only action you can take is to project your soul up to 100 feet out of the container, either returning to your living body (and ending the spell) or attempting to possess a humanoids body.
On the follow up:
Is there a difference...?
Technically? Yes, I believe so. Mechanically? Not as far as I can tell.
From both a linguistics and RAW perspective, "can't move" sounds like "cannot take the move portion of their turn", as in the PHB p. 189:
On your turn, you can move a distance up to your speed and take one action.
While "speed becomes 0, and it can't benefit from any bonus to its speed." sounds like you can use your move, but you can only go zero feet and nothing can increase that distance.
The only case I can imagine where these would not yield the same in-game results is if there were a way to use your move to cause an effect while still spending no more than zero feet of your movement. I have yet to find such a case. There are spells that spend feet of movement like currency, such as Tree Stride (PHB p.283), but I have found no instances where one can cause an effect using 0 feet of movement.
They aren't different in the sense of "tactical movement", but they're very different concepts from the narrative perspective.
If your speed is reduced to zero, you can't move around (i.e. change which square you're standing in), but you can still move your body around, like waving your arms or looking around.
If you can't move, you literally can't move; you've been frozen in place or gone limp. Your arms and legs aren't responding, you can't turn your head, nothing is working.
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:
In each of these conditions there is additional description indicating that, along with any of those conditions, a creature is also Incapacitated:
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:
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).
I think 'can't move' means you literally can't physically move at all, so you can't struggle or attack.
Whereas 'speed becomes zero' means you can't Move in the technical sense, but you can still attack, attempt to break free etc.
A speed of 0 means that you can't make use of your movement abilities eg. walking, flying, climbing, burrowing. The move action lets you move a distance up to your speed. You can use the dash action to increase the distance you can move via your speed. Having a speed of 0 doesn't mean you can't move at all though, you just can't do it via a move action. You can use methods such as spells like dimension door or misty step. If you can't move however, then you can't move under your own power no matter what method you might try. This also includes not being able to talk, attack, use items etc.
