The sphere is weightless, even when filled
Although the spell could certainly be more clearly written, the language context clues tell us that not only is the sphere itself weightless, but that it renders the occupant effectively weightless for the purpose of being moved by others.
The sphere is weightless and just large enough to contain the creature or object inside. An enclosed creature can use its action to push against the sphere's walls and thus roll the sphere at up to half the creature's speed. Similarly, the globe can be picked up and moved by other creatures. [Actual spell text]
Let's consider this last sentence, especially the word 'similarly'. Suppose the sphere itself was weightless, but a creature wanting to move it had to contend against the weight of the creature inside, using the normal rules for lifting and carrying. If that were the case, it could simply say, "The globe can be moved by other creatures." Or, more explicitly, "The globe can be picked up and moved by other creatures according to the weight of the creature inside." But it does not say that. Rather, it adds the word "similarly" - so what about the last sentence is similar to the one preceding it?
Certainly not who is moving it; that is clearly different. Nor how it is being moved - it is being picked up, not pushed. Thus the only things that could be similar are the cost to move it (an action) and the speed at which it can be moved (half the speed of the mover). Therefor, what the last two sentences are actually saying is:
An enclosed creature can use its action to push against the sphere's walls and thus roll the sphere at up to half the creature's speed. Another creature can also use its action, but to pick up and move the sphere at up to half that creature's speed. [interpretation of intent]
The spell is saying that it is granting the ability to move the creature inside to any creature on the outside that picks up the sphere, and that this is a special situation created by the spell. If the normal rules of lifting and carrying applied, why say that a creature on the outside can lift the sphere? It would be as redundant as if the conjure animals spell said, "By the way, you can pick up the beasts you summon if you are strong enough." Of course you can!. To pick up the sphere while taking into account the weight of the creature inside, however, you would need to know its weight - which you almost never would, since NPC stat blocks contain size, not weight.
Further, even knowing the weight of the creature inside the sphere, you would often not be able to pick the creature up at all, since the total weight you can move is twice your carrying capacity (including the gear you already have on), and at anything more than your carrying capacity you can move at only 5 feet, not half. You can also move a creature by grappling and dragging it at half movement, but you are limited to creatures of one size larger than yourself. Since the sphere can enclose a creature of size Large, by means of the spell, even a size Small PC can pick up and move a size Large creature.
The default assumption of D&D is that the objects and creatures created by spells interact with the world the way you would expect them to, unless the spell description says otherwise. By specifying that the sphere can be picked up for an action and moved at half our movement, the spell is telling us this is unusual behavior, unusual because it is possible regardless of the trapped creature's weight, as if it were stuck in some sort of cartoon soap bubble that somehow buoys it up. The spell says that the sphere is weightless - even when there is a creature inside, which there almost always will be, since "the creature is enclosed for the duration."
[Note that in comments, Matthieu suggests that the special nature of the sphere is not that it renders its captive weightless, but that it can be moved at all, since other spells that create force barriers are immobile. This is a valid point, but if we extend our comparison to magic items, we see that a cube of force is mobile as well.]
If we accept that the purpose of the spell is to make its captive weightless and transportable, ]other parts of the spell become more clear:
The sphere is immune to all damage, and a creature or object inside can't be damaged by attacks or effects originating from outside, nor can a creature inside the sphere damage anything outside it.
The creature inside can't be damaged by effects originating from outside - and this would include falling damage. Drop your ensphered creature off a cliff and it might plummet, or might gently drift down, but it won't take damage when it hits the bottom, and it won't do damage to anything underneath it when it hits - and that only makes sense if not only the sphere is weightless, but if it renders null the weight of the creature inside as well.
Nothing--not physical objects, energy, or other spell effects--can pass through the barrier, in or out, though a creature in the sphere can breathe there.
D&D is not a physics engine, so we won't quibble about the differences between energy and force, but suffice to say 'nothing can pass through' includes gravity going in, and the weight of the creature going out, which is the reason the weight of the creature inside does not get translated to anyone outside picking up the sphere.