You do not need to carry it
Both versions of the cannon must be initially placed in an unoccupied space, so it must be true that they can exist without being carried. Why then, the distinction? Well, the wording is simply giving each version of the cannon a piece of rules text that is specific to it.
A Small eldritch cannon occupies its space (and a Tiny one does not)
A space in which a small cannot resides is considered "occupied." Effects that require an unoccupied space cannot make use of that area. It's not clear whether hostile creatures can move through the space, but since the space is "occupied," it's likely that no creature can end its turn in that space. This cannon might be used as partial cover or to prevent flanking, depending on the situation.
The only thing implied by making this descriptor specific to the Small cannon is that it is not true of the Tiny cannon. The Tiny cannon does not occupy its space, and that space can be used for any other feature that needs it. It's clear that the space can be moved through by any creature.
A Tiny one can be held in one hand (and a Small one cannot)
A Tiny cannon is small enough to hold in only one hand. You can carry it around while using your other hand for whatever you'd like.
The only thing implied by making this descriptor specific to the Tiny cannon is that it is not true of the Small cannon. The Small cannon cannot be held in one hand, and thus requires two to carry or move.