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Originally Posted by citivolus
As for the shape being a sphere rather than a plane, that would be due to the diminishing impact of the Sun's gravity on the objects as distance is increased. The closer objects in the cloud would like closer to the plane of the solar system, while further objects would be more spread out due to the higher ratio of gravity from extra-solar objects impacting their orbits. It is postulated that all the objects originated in the solar plane, but that the outer ones have migrated in their orbits over time due to those extra-solar forces. The overall shape of the positional probability of the objects would probably look more like two balloons pressed together inside another balloon than a sphere.
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The disc-like shape of the cloud closer to the Sun from which the planets are formed is the consequence of the preservation of orbital momentum of the primordial cloud as a whole.. After collapse due to gravity, the material has a tendency to concentrate in the disk simply because the momentum of the whole cloud is preserved this way..
The Oort cloud is not collapsed so it has its original shape, that is very roughly spherical. The perturbation of the nearby stars as well as tidal forces of the Milky way spreads it out additionally.