Omni (all) direction means "in every direction."
The importance of omnidirection in tensegrity
Structure emerges from the deployment of tension and compression in proper ratio. A Roman Arch from the time of Jesus deplys both tension and compression as surely as does Snelson's Needle Tower. The difference is in the omnidirectional nature of the tensional network.
In a Roman Arch, the arch's stones and keystone rely on gravitation providing constant acceleratory force, or tension, in one direction only: from the keystone, between the arches posts, to the Earth. If gravity were to suddenly shift, or the arch turned upside down, the stones would no longer cohere in a structure.
In Snelson's Needle Tower, by contrast, the deployment of tension and compression in the structure are independent of the direction of gravity. The tower can be turned upside-down without changing its structure.
When the Roman Arch technique is deployed in a round, barrel format, the closed cylinder described by the compressional staves is held together by the tensional hoops surrounding the barrel. The resulting structure is omnidirectionally cohesive, and so displays the properties of a tensegrity.
Omnidirection as a distinguishing factor between tensional structures
Many structures such as suspension bridges and hyperbolic fabric tents deploy tension as a structural component. However, these structures do not have omnidirectional tensional integrity. That is one important reason why they are not considered tensegrities.